On a Heegaard Floer theory for
tangles
Claudius Bodo Zibrowius
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
This dissertation is submitted for the degree of
Doctor of Philosophy
Trinity Hall March 2017

iii

Declaration
This dissertation is the result of my own work and includes nothing which
is the outcome of work done in collaboration except as declared in the
acknowledgements and specified in the text.
It is not substantially the same as any that I have submitted, or is being
concurrently submitted, for a degree or diploma or other qualification at the
University of Cambridge or any other University or similar institution except
as declared in the acknowledgements and specified in the text. I further
state that no substantial part of my dissertation has already been submitted,
or is being concurrently submitted, for any such degree, diploma or other
qualification at the University of Cambridge or any other University or
similar institution except as declared in the acknowledgements and specified
in the text.
Claudius Zibrowius
March 2017
v

Abstract
The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s
Heegaard Floer homology ĤFL for links in the 3-sphere, i. e. a Heegaard Floer
homology ĤFT for tangles in the 3-ball.
The decategorification of ĤFL is the classical Alexander polynomial for links;
likewise, the decategorification of ĤFT gives a local version ∇sT of the Alex-
ander polynomial. In the first chapter of this thesis, we give a purely combi-
natorial definition of this polynomial invariant ∇sT via Kauffman states and
Alexander codes and investigate some of its properties. As an application, we
show that the multivariate Alexander polynomial is mutation invariant.
In the second chapter, we define ĤFT in two slightly different, but equivalent
ways: One is via Juhász’s sutured Floer homology, the other by imitating the
construction of ĤFL. We then state a glueing theorem in terms of Zarev’s
bordered sutured Floer homology, which endows ĤFT with additional struc-
ture. As an application, we show that any two links related by mutation
about a (2,−3)-pretzel tangle have the same δ-graded link Floer homology.
This result relies on a computer calculation.
In the third and last chapter, we specialise to 4-ended tangles. In this case,
we give a reformulation of ĤFT with a glueing structure in terms of (what
we call) peculiar modules. Together with a glueing theorem, we can easily
recover oriented and unoriented skein relations for ĤFL. Our peculiar mod-
ules also enjoy some symmetry relations, which support a conjecture about
δ-graded mutation invariance of ĤFL. However, stronger symmetries would
be needed to actually prove this conjecture. Finally, we explore the relation-
ship between peculiar modules and twisted complexes in the wrapped Fukaya
category of the 4-punctured sphere.
There are four appendices, some of which might be of independent interest:
In the first appendix, we describe a general construction of dg categories
which unifies all algebraic structures used in this thesis, in particular type A
and type D modules from bordered theory. In the second appendix, we prove
a generalised version of Kauffman’s clock theorem, which plays a major role
for our decategorified invariants. The last two appendices are manuals for
two Mathematica programs. The first is a tool for computing the generators
of ĤFT and the decategorified tangle invariant ∇sT . The second allows us to
compute bordered sutured Floer homology using nice diagrams.
vii

Acknowledgements
First and foremost, I would like to thank my supervisor Jake Rasmussen for
his generous support throughout the entire time of my PhD and before, dur-
ing Part III. I consider myself very fortunate to have been his student.
My PhD was funded by an EPSRC scholarship covering tuition fees and a
DPMMS grant for maintenance, for which I thank the then Head of Depart-
ment Martin Hyland. I also gratefully acknowledge a research studentship
from the Cambridge Philosophical Society for Michaelmas Term 2016.
I thank my examiners Ivan Smith and András Juhász for many valuable
comments on and corrections to the soft-bound version of this thesis that I
prepared for my viva. I also thank Mohammed Abouzaid, Guillem Cazassus,
Celeste Damiani, Artem Kotelskiy, Peter Lambert-Cole, Adam Levine, Ina
Petkova, Vera Vértesi and my brother Marcus Zibrowius for helpful conver-
sations. My special thanks go to Liam Watson for his interest in my work,
his support and the opportunity to speak in Glasgow twice.
I am very grateful to my PhD brothers Tom Brown, Tom Gillespie and Paul
Wedrich, and my fellow PhD students Nina Friedrich and Christian Lund
for their company and friendship. I would also like to thank Senja Barthel,
Fyodor Gainullin, Tom Hockenhull and Marco Marengon for organising yearly
student conferences at Imperial College London, all of which were terrific.
I thank Johnny Nicholson for helpful comments on an earlier draft of chapter I
and for sharing his computations with me during an undergraduate summer
research project in 2016.
I am indebted to Tom Brown, Nina Friedrich, Paul Wedrich, Marcus
Zibrowius and, especially, my father for their proof-reading services.
No line of this thesis would have been written without the love and support
of my brother and parents. This thesis is dedicated to them.
ix

Table of Contents
Dedication iii
Declaration v
Abstract vii
Acknowledgements ix
Introduction 1
Chapter I. Alexander polynomials for tangles 9
I.1. The tangle invariant ∇sT 10
I.2. Basic properties of ∇sT 15
I.3. 4-ended tangles and mutation invariance 22
I.4. Geometric interpretation of ∇sT 26
Chapter II. A Heegaard Floer homology for tangles 33
II.1. A categorification via sutured Heegaard Floer theory 33
II.2. Heegaard diagrams for tangles 37
II.3. Glueing via bordered sutured Floer theory 46
Chapter III. Peculiar invariants for 4-ended tangles 59
III.1. A glueing structure on ĤFT for 4-ended tangles 59
III.2. Pairing 4-ended tangles 71
III.3. Skein relations 80
III.4. Symmetry relations for CFT∂ 84
III.5. CFT∂ and the wrapped Fukaya category of the 4-punctured sphere 87
Appendix A. Algebraic structures from dg categories 97
Appendix B. Proof of the generalised clock theorem 105
Appendix C. Manual for APT.m 111
Appendix D. Manual for BSFH.m 115
Bibliography 121
xi

Introduction
Let L be a link in the 3-sphere S3. Consider a closed 3-ball B3 ⊂ S3 whose boundary
intersects L transversely. Then L ∩ B3 is essentially what we call a tangle, the main pro-
tagonist of this thesis. We define a tangle invariant ĤFT, a Heegaard Floer homology for
tangles, and study its properties.
Heegaard Floer homology theories were first defined by Ozsváth and Szabó in 2001 [OS01].
With an oriented, closed 3-dimensional manifold M , they associated a family of homologi-
cal invariants, the simplest of which is denoted by ĤF(M). Given an oriented knot in M ,
Ozsváth and Szabó, and independently Rasmussen, then defined filtrations on the chain
complexes which give rise to the respective flavours of knot Floer homology [OS03a, Ras03].
This was later generalised to oriented links in S3 [OS05]. Our tangle Floer homology should
be understood as a generalisation of the hat version of link Floer homology ĤFL to oriented
tangles.
ĤFT and an Alexander polynomial for tangles. Like ĤFL, our tangle Floer ho-
mology is a finitely generated Abelian group which comes with two gradings: a relative
homological Z-grading and an Alexander grading, which is an additional relative Z-grading
for each component of the tangle. However, unlike ĤFL, our tangle Floer homology depends
on some extra data, a site, associated with a tangle, see definition I.1.6. For a tangle T with
n open strands, there are
(
2n
n−1
)
such sites s, and for each of them, we define a bigraded chain
complex
ĈFT(T, s) =
⊕
h∈Z ←homological grading
a∈Z|T | ←Alexander grading
ĈFTh(T, s, a),
where |T | denotes the number of components of T .
Theorem 0.1 (II.2.22). Given a tangle T and a site s for T , the bigraded chain homotopy
type of ĈFT(T, s) is an invariant of T . We denote its homology by ĤFT(T, s) and call it
the tangle Floer homology of T .
In link Floer homology, the “Alexander” in “Alexander grading” comes from the fact
that, given a link L in S3, the graded Euler characteristic of ĤFL(L) recovers the Alexander
polynomial of L, a classical polynomial link invariant, named after its discoverer [Ale28].
We say link Floer homology categorifies the Alexander polynomial. Similarly, for tangles,
we obtain polynomial invariants
χ(ĤFT(T, s)) =
∑
h,a
(−1)h rk(ĤFTh(T, s, a)) · ta11 · · · t
a|T |
|T | ∈ Z[t±11 , . . . , t±1|T |],
1
2 INTRODUCTION
which are well-defined up to multiplication by a unit. In chapter I, we give a purely com-
binatorial definition of a normalised version ∇sT of these polynomial invariants in terms of
Kauffman states and Alexander codes and study their properties.
Mutation. A new invariant can already be interesting because of its simplicity or its
aesthetic appeal. But its true value should be determined by its capacity to answer questions
about existing theory. So the primary purpose of any tangle Floer homology should be to
learn more about “the local nature” of knot and link Floer homology and, ultimately, of
the geometric objects themselves. A prime example of an open question one might hope to
address is how link Floer homology behaves under mutation.
Definition 0.2. Let L be a link. Construct a new link L′ by cutting out a 4-ended
tangle Γ and glueing it back in after a half-rotation, as illustrated below:
Γ −→ Γ
We say L′ is obtained from L by Conway mutation. We call Γ the mutating tangle and
L′ a mutant of L. If L is oriented, we define an orientation on L′ such that it agrees with
the one on L outside Γ. If this means that we need to reverse the orientation of the two open
components of Γ, we also reverse the orientation of any closed components of Γ; otherwise,
we do not change the orientation on Γ.
We know from [OS03b] that knot and link Floer homology is, in general, not invariant
under mutation. However, we have the following conjecture from [BL11, conjecture 1.5].
Conjecture 0.3. Let L be a link and let L′ be obtained from L by Conway mutation.
Then ĤFL(L) and ĤFL(L′) agree after collapsing the bigrading to a single Z-grading, known
as the δ-grading. In short: δ-graded link Floer homology is mutation invariant.
Let us see what the new invariants tell us on the decategorified level, i. e. on the level of
the Alexander polynomial.
Theorem 0.4 (I.3.6). The multivariate Alexander polynomial is invariant under Conway
mutation after identifying the variables corresponding to the two open strands of the mutating
tangle.
This result has long been known for the single-variate Alexander polynomial, see for
example [LM87, proposition 11], but I have been unable to find a result for the multivariate
polynomial in the literature. The proof of theorem 0.4 relies on certain symmetry relations
between the invariants ∇sT for 4-ended tangles T and varying sites s, see proposition I.3.1.
For ĤFT, we can prove similar symmetry relations which categorify those for ∇sT . As
conjecture 0.3 suggests, in general, they only hold for δ-graded tangle Floer homology, see
proposition II.1.8 and example II.2.26. However, these symmetry relations are not sufficient
to prove the conjecture. This is because, unlike ∇sT , ĤFT alone is insufficient to state a
glueing formula.
INTRODUCTION 3
Glueing tangle Floer homologies. The main tool for glueing 3-manifolds with bound-
ary in Heegaard Floer homology is bordered Heegaard Floer homology, developed by Lip-
shitz, Ozsváth and Thurston [LOT08]. In [Zar09], Zarev generalised this theory to sutured
manifolds. We interpret our tangle Floer homology ĤFT in terms of Zarev’s theory to add
a glueing structure to ĤFT. This glueing structure essentially takes the form of extra dif-
ferentials between the tangle Floer chain complexes ĈFT(T, s) for different sites s. As an
accompaniment to this thesis, we provide the Mathematica package [BSFH.m] which allows
us to compute the bordered sutured invariants for any bordered sutured manifold from nice
diagrams, see appendix D for a documentation of [BSFH.m]. In particular, it allows us to
confirm conjecture 0.3 for mutation about a particular non-trivial tangle.
Theorem 0.5 (II.3.14, III.1.18). Consider the following (2,−3)-pretzel tangle:
Two knots or links that are related by mutation of this tangle have the same bigraded knot or
link Floer homologies after identifying the Alexander gradings corresponding to the two open
strands. If the orientation of one of those two strands is reversed, then their δ-graded knot
or link Floer homologies agree.
We offer two independent proofs of this result, both of which, however, rely on calcu-
lations using the program [BSFH.m]. The first proof follows from computing the glueing
structure for the (2,−3)-pretzel tangle in two different ways corresponding to mutation and
observing that the two results are homotopy equivalent, see theorem II.3.14. However, this
proof does not really tell us why they are homotopic. The second proof answers this ques-
tion more satisfactorily, replacing the previous ad hoc construction by a conceptually more
refined approach, see example III.1.18 in conjunction with theorem III.2.5. To explain this,
let us discuss bordered invariants in a little more detail.
Bordered theory and Fukaya categories. In general, the invariants and the glue-
ing theorems in bordered and also bordered sutured Heegaard Floer homology look rather
complicated. With a closed 3-manifold M split along a (parametrised) closed surface F into
two components M1 and M2, one associates a differential graded algebra A(F ), a so-called
type D module ĈFD(M1) and a type A module ĈFA(M2) over A(F ). Then ĤF(M) can be
computed as the homology of a special tensor product  of ĈFD(M1) and ĈFA(M2) over
A(F ). If one wants to split M into more than two pieces, there are also bimodule invariants
of type AA, AD, DA and DD, depending on whether we treat the glueing surfaces of the
components as type A or type D sides.
Especially the algebra A(F ) can be quite complicated, and in general, it will be so for our
tangle Floer homology. In [Aur10], Auroux gave an interpretation of the bordered algebra
4 INTRODUCTION
A(F ) in terms of some partially wrapped Fukaya category and outlined a conjectural re-
formulation of bordered Heegaard Floer theory in this framework. In [LOT10], Lipshitz,
Ozsváth and Thurston gave further evidence for this conjectural relationship by restating
their glueing theorem purely in terms of the homology of the space of morphisms between
the two type D modules of M1 and M2. Moreover, Rasmussen, Hanselman and Watson very
recently gave an elegant reformulation of the glueing theorem for a surprisingly large class of
manifolds with torus boundary, which they called loop-type [HRW16]. For such manifolds,
the type D structure can be interpreted as an object in the Fukaya category of immersed
curves on a (punctured) torus. In particular, glueing corresponds to taking Lagrangian in-
tersection homology on the torus.
For 4-ended tangles, a similar story seems to be true, which we explore in chapter III.
A glueing structure for 4-ended tangles.
Theorem 0.6 (III.1.11). Given a 4-ended tangle T , we can endow the tangle Floer ho-
mology ĤFT with an additional structure of a curved complex, which we denote by CFT∂(T ).
(Roughly speaking, a curved complex is a type D module for which the differential does not
square to 0, see definition A.11.) CFT∂(T ) is a tangle invariant. We call it the peculiar
module of T .
This enables us to explicitly calculate objects for 4-ended tangles in the (triangulated
enlargement of the) fully wrapped Fukaya category TwFuk(S2, 4) of the 4-punctured sphere,
via the A∞-functor L in the following theorem.
Theorem 0.7 (III.5.10). Let pqMod be the category of peculiar modules. There exist
two non-trivial A∞-functors
pqMod TwFuk(S2, 4).
L
M
This allows us to interpret the sites of 4-ended tangles explicitly in terms of generators
of the Fukaya category TwFuk(S2, 4) via the following proposition.
Proposition 0.8 (III.5.15). Let T be a 4-ended tangle and s a site of T . Then, there
exists a generator Ls of TwFuk(S2, 4) such that ĈFT(T, s) is bigraded chain homotopic to
the Lagrangian intersection chain complex(
Mor
(
Ls,L
(
CFT∂(T )
))
, µtw1
)
,
where µtw1 denotes the differential on morphism spaces in TwFuk(S
2, 4).
We expect that one can extend the proof of the result above to show the following.
Conjecture 0.9 (III.5.14). The functors M and L from theorem 0.7 define an equiv-
alence of A∞-categories.
For rational tangles, but also for more complicated ones like the (2,−3)-pretzel tangle,
the peculiar modules are loop-type in a similar sense to [HRW16]: the tangle Floer homology
of these tangles is represented by a collection of loops in the 4-punctured 2-sphere, see figure 1
and examples III.1.17 and III.1.18. Furthermore, computations suggest that we can indeed
compute link Floer homology as the Lagrangian intersection homology of such immersed
curves. In fact, we have the following glueing result.
INTRODUCTION 5
b
b
bb
b
b
(a)
b
b
b
b
b
b
(b)
b
b
b b
b
b
(c)
Figure 1. The three loops on the 4-punctured sphere respresenting CFT∂ for the
(2,−3)-pretzel tangle of theorem 0.5. The intersection points of the loops with the
dashed arcs connecting the four punctures calculate the underlying non-glueable
tangle Floer homology ĤFT. The splitting of ĤFT according to the four different
sites is indicated by the colouring of the intersection points.
Theorem 0.10 (III.2.1 and III.2.5). Let T1 and T2 be two 4-ended tangles and L the
link obtained by glueing them together according to the following picture.
T1 T2
Then there exists a certain type AA bimodule P such that ĈFL(L) is equal to
CFT∂(T1) P  CFT∂(T2)
up to at most three stabilisations, i. e. tensoring with a certain 2-dimensional vector space.
Furthermore, for loop-type CFT∂(T1) and trivial T2, ĈFL(L) agrees with the Lagrangian
intersection Floer homology of the loop of T1 with that of the trivial tangle, up to at most a
single stabilisation.
The computation of the type AA bimodule P is done using [BSFH.m]. The second
statement follows from computing a type A structure, which can be done by hand, or al-
ternatively, by simplifying P  CFT∂(T2). We expect the second statement to generalise
to pairings of arbitrary (loop-type) tangles. Unfortunately, however, the bimodule P looks
rather complicated and not like the one to be expected from the Fukaya category. So there
remains work to be done, see conjectures III.2.7 and III.5.16.
Nonetheless, the mere existence of a glueing theorem for CFT∂ allows us to infer properties
of link Floer homology. For example, the peculiar invariant of the (2,−3)-pretzel tangle
has an intrinsic symmetry (see figure 1), which gives us the second proof of theorem 0.5.
We expect other 2-stranded pretzel tangles to have the same kind of symmetry; however,
a slightly more careful analysis of holomorphic curves is needed to determine the structure
maps.
Conjecture 0.11. Theorem 0.5 generalises to any 2-stranded pretzel-tangle.
6 INTRODUCTION
n

...
(a) Tn
n

...
(b) T−n (c) T0
Figure 2. The basic tangles for the skein exact sequence from theorem 0.13
For general 4-ended tangles, we are only able to prove slightly weaker symmetry relations
for CFT∂ . They support the mutation conjecture, see propositions III.4.1 and III.4.2, but
they do not seem to be sufficient to prove it.
As another application of the existence of a glueing theorem, we show that peculiar modules
detect rational tangles:
Theorem 0.12 (III.2.8). A 4-ended tangle T is rational iff CFT∂(T ) is homotopic to a
single loop that corresponds to an embedded loop on the 4-punctured sphere.
Furthermore, we can easily reprove the existence of an unoriented skein exact sequence
[Man06], see theorem III.3.3. Similarly, we obtain the following slight generalisation of
Ozsváth and Szabó’s oriented skein exact sequence [OS03a].
Theorem 0.13 (III.3.1, see also III.3.2). Let Tn be the positive n-twist tangle, T−n the
negative n-twist tangle and T0 the trivial tangle, see figure 2. Then there is an exact triangle:
CFT∂(T−n) CFT∂(Tn)
CFT∂(T0)⊗ V
where V is some 2-dimensional vector space. If the tangles are oriented and coloured con-
sistently, one obtains (bi)graded versions of this triangle. Furthermore, it gives rise to an
exact triangle relating the (appropriately stabilised) link Floer homologies of links that differ
in these three tangles.
Parallels to Khovanov homology. Our peculiar invariants CFT∂ are also interesting
from another, perhaps more philosophical perspective, namely the relationship between link
Floer homology and Khovanov homology. Khovanov homology is another homology theory
for knots and links, first defined by Khovanov in 1999 [Kho99]. It categorifies the Jones
polynomial in the same way that ĤFL categorifies the Alexander polynomial. Although the
two theories are defined and computed in very different ways, they look quite similar from
a formal point of view, see for example [Ras05].
In [Bar04], Bar-Natan gave an elegant generalisation of Khovanov homology to tangles. With
a tangle diagram, he associated an (up to homotopy) invariant chain complex over a certain
category which essentially (that is, up to grading) consists of only finitely many objects
and morphisms. For example, for 4-ended tangles, there are just two objects and at most
four morphisms in each hom-set. For CFT∂ , we similarly get two candidates for such basic
INTRODUCTION 7
Khovanov homologyknot Floer homology
Alexander polynomial Jones polynomial
homology for tangles
Bar-Natan’s Khovanov
?
? for tangles
Jones polynomial
categorifiescategorifies
categorifies
Figure 3. Another perspective on ∇sT , ĤFT and especially CFT∂
objects, and we sketch how to write the peculiar module of any tangle as a chain complex
in these basic objects, up to a large tensor factor, see remark III.3.5. For the (2,−3)-pretzel
tangle, this tensor factor can be removed.
Questions 0.14. Given any oriented 4-ended tangle T , can we write CFT∂(T ) as a
chain complex in two basic objects? If so, can we describe the chain maps?
An affirmative answer to these two questions would not only be aesthetically pleasing.
The basic objects are symmetric under mutation, so if the chain maps between the basic
complexes also have this symmetry, one might be able to prove the mutation conjecture.
In Khovanov homology, such an approach has been successful: Khovanov homology with
Z/2-coefficients is known to be mutation invariant, see [Weh09].
Similar work by other people. There are several other groups of people working on
similar ideas to those described in this thesis.
In 2014, Petkova and Vértesi defined a combinatorial tangle Floer homology using grid dia-
grams and ideas from bordered Floer homology [PV14]. They use a more general definition
of tangles, namely those with a “top” and a “bottom”, i. e. braids with caps and cups. In
[EPV15], they and Ellis show that the decategorification of their invariant agrees with Sar-
tori’s generalisation of the Alexander polynomials to top-bottom-tangles via representations
of Uq(gl(1|1)) [Srt13]. Thus, Petkova and Vértesi’s theory fits very nicely into sln-homology
theories arising from Khovanov homology.
Very recently, Ozsváth and Szabó developed a completely algebraically defined knot homol-
ogy theory, which they conjecture to be equivalent to knot Floer homology [OS16]. Like
Petkova and Vértesi, they cut up a knot diagram into elementary pieces, so they automat-
ically obtain tangle invariants, too. This theory seems to be frightfully powerful: from a
computational point of view, since they can compute their homology from diagrams with
over 50 crossings; but also from a more theoretical point of view, since their theory includes
the hat- as well as the more sophisticated “−”-version of knot Floer homology without ref-
erence to holomorphic curves or grid diagrams. Interestingly, the generators in their theory
correspond to Kauffman states like in ours.
8 INTRODUCTION
In [HHK13, HHK15], Hedden, Herald and Kirk study the Lagrangian intersection homology
of immersed curves on a 4-punctured sphere (the “pillowcase”) in the context of instanton
knot Floer homology, a computationally more difficult knot Floer homology due to Kron-
heimer and Mrowka, which also categorifies the Alexander polynomial and is conjecturally
closely related to ĤFK [KM10]. The curve they associate with the trivial tangle does not
agree with ours, but it looks very similar to the curve we associate with a singular crossing,
see proposition III.3.4 and remark III.3.5.
Finally, I want to mention recent work of Lambert-Cole on conjecture 0.3. In [L16], he shows
that the bigraded knot Floer homologies of a mutant knot pair obtained by introducing a
sufficiently large number of twists into a given (positive) mutant knot pair agree. This follows
from a certain stabilisation property of knot Floer homology with respect to twisting that
he proves in the same paper. In [L17], he investigates conjecture 0.3 from a more axiomatic
point of view, using basepoint maps and Manolescu’s unoriented skein exact triangle as basic
ingredients. He is able to confirm conjecture 0.3 for mutating tangles that can be closed to
an unlink by a rational tangle. In particular, this result encompasses the δ-graded part of
theorem 0.5.
Outline. The thesis is split into three chapters, following not only a logical, but inci-
dentally also a roughly chronological order.
The first chapter is purely concerned with the decategorified story, the combinatorial defini-
tion of the polynomial tangle invariants ∇sT and their properties. The chapter can be seen
as a playground for testing ideas for chapters II and III. In fact, some properties of ∇sT are
immediate consequences of their categorified counterparts. However, other questions, most
prominently the one concerning mutation invariance, are still unresolved in the categorified
setting, whilst being relatively easy to answer for the decategorified invariants. On a first
read, one can skip all sections of this chapter except the very first.
In chapter II, we define the tangle Floer homology ĤFT; first, via sutured Floer homology,
then in more detail via Heegaard diagrams for tangles, imitating the definition of link Floer
homology. We then interpret our invariant in terms of Zarev’s bordered sutured theory.
In the third and final chapter, we specialise to 4-ended tangles and repackage the glueing
structure in this special case into the peculiar invariant CFT∂ . We investigate some of its
properties and discuss several applications and open questions.
There are four appendices. In appendix A, we describe the algebraic structures appearing
in this thesis from an abstract category-theoretic point of view and derive useful tools for
working with them, which form the basis of all our computations. In the second appen-
dix, we give a proof of the generalised clock theorem, which is essential for studying the
polynomial invariants ∇sT in chapter I. The last two appendices are documentations for the
programs [APT.m] and [BSFH.m].
CHAPTER I
Alexander polynomials for tangles
In the first chapter, we define and study the polynomial invariant ∇sT , a generalisation
of the Alexander polynomial of knots and links to tangles. The Alexander polynomial is a
classical knot and link invariant which takes the form of a Laurent polynomial in the same
number of variables as there are link components [Ale28]. We will use its normalised form,
which is also known as the Conway potential function and denoted by ∇L, see for example
Hartley’s monograph [Har83]. This polynomial invariant can be defined and interpreted in
many different ways, depending on one’s preferred point of view. For example, this might
be Fox calculus, elementary ideals, Reidemeister torsion or skein theory.
We start from Kauffman’s combinatorial definition of the Conway potential function for
knots and links. In section I.1, we adapt this definition to tangles. In general, this gives us
a finite set of Laurent polynomials ∇sT associated with an oriented tangle diagram T . This
finite set of invariants is indexed by some additional input data for tangles, which we call
the sites of T (see definition 1.8). In theorems 1.11 and 1.12, we show the following basic
result.
Theorem. For each site s of an oriented tangle T , ∇sT is an invariant of T . Further-
more, if T represents a link or a knot L, there is exactly one site s, namely s = ∅, and ∇∅T
is equal to the Conway potential function ∇L up to a certain factor.
The definition of ∇sT is a very straightforward generalisation of Kauffman’s construction.
However, apart from a short discussion in [GL86], I am unaware of any reference in the
literature where this invariant has been studied. So this will be object of the remaining part
of this chapter.
We show that the invariants∇sT satisfy a glueing formula (proposition 1.15) which generalises
the connected sum formula for the knot and link case. In section I.2, we derive some
further properties of our tangle invariants. As we will see, many properties of the Conway
potential function for knots and links generalise. In particular, they are well-behaved under
orientation reversal of the tangle strands and taking mirror images. In section I.3, we
study ∇sT for 4-ended tangles and prove symmetry relations between different sites s. They
imply that ∇sT and in particular also the multivariate Alexander polynomial of links is
mutation invariant, for a precise statement, see theorem 3.6. In section I.4, we interpret
the invariants geometrically in terms of the first homology of the maximal Abelian cover of
the tangle complement relative to certain subspaces of its boundary. Because this geometric
generalisation of the Alexander polynomial looks even more natural than the one using
Kauffman states (but with the slight drawback of being unnormalised), this seems to be a
good starting point for any comparison with other constructions of Alexander polynomials
for tangles, of which there are many [Arc10, Pol10, Big12, Ken12, BCF12, Srt13, DV16].
9
10 I. ALEXANDER POLYNOMIALS FOR TANGLES
I.1. The tangle invariant ∇sT
First of all, we define what we mean by a tangle. Our definition is based on Conway’s
notion of tangles, see for example [Ada94, section 2.3].
Definition 1.1. A tangle T is an embedding of a disjoint union of intervals and circles
into the closed 3-ball B3,
T :
(∐
I q
∐
S1, ∂
)
↪→ (B3, S1 ⊂ ∂B3) ,
such that the endpoints of the intervals lie on a fixed circle S1 on the boundary of B3,
together with a labelling of the arcs S1r im(T ) by some index set {a, b, c, . . . }. We consider
tangles up to ambient isotopy which keeps track of the labelling of the arcs. If the number
of intervals is n, we call a tangle 2n-ended. The images of the intervals are called open
components, the images of the circles are called closed components. We often label
these tangle components by variables p, q, r, . . . or t1, t2, . . . , which we call the colours of T .
In analogy to link diagrams, we define a tangle diagram D to be an immersion of intervals
and circles into the closed 2-disc,
D :
(∐
I q
∐
S1, ∂
)
#
(
D2, ∂D2
)
,
whose image is a graph with 1- and 4-valent vertices only, together with under/over infor-
mation at each 4-valent vertex and a labelling of the arcs ∂D2 rD(∂I) by some index set
{a, b, c, . . . }. Just as in the case of links, we consider tangle diagrams up to ambient isotopy
and the usual Reidemeister moves, see for example [Lic97]. Connected components of the
complement of the image of D are called regions. Those regions that meet ∂D2 are called
open, the others are called closed. We call a diagram connected if the intersection of each
open region with ∂D2 is connected. Unless specified otherwise, diagrams are assumed to be
connected and have at least one crossing.
Remark 1.2. Regard D2 as the intersection of B3 with the plane {z = 0}. Given a
tangle diagram D, we can remove the singularities of the immersion D by pushing the two
components at each singularity into {z > 0} and {z < 0}, according to the under/over
information. Conversely, given a tangle T , we can choose an embedded disc D2 bounding
the fixed circle S1. Then, just as in the case of links, a generic projection of B3 onto this
disc gives rise to a well-defined tangle diagram. So in the following, we use “tangles” and
“tangle diagrams” synonymously, unless it is clear from the context that we do not.
Lemma 1.3. Let T1 and T2 be two oriented connected tangle diagrams that represent the
same tangle. Then there is a sequence of Reidemeister moves that connects T1 to T2 via
connected diagrams.
Proof. By the previous remark, we can always find a sequence of Reidemeister moves
connecting the two diagrams. To ensure that all diagrams are connected, we pick one open
strand near the boundary and pull it once around the whole diagram, using Reidemeister II
moves before we go along the sequence of Reidemeister moves and undo the first step once
we have arrived at T2. 
I.1. THE TANGLE INVARIANT ∇ST 11
a
b
c
d
(a)
a
b
c
d
(b)
a
b
c
d
(c)
a
b
c
d
(d)
a
b
c
d
(e)
Figure 4. Some diagrams of rational tangles. (a) and (b) represent the same tangle,
but only (b) is a connected diagram. (c) and (d) show some more complicated rational
tangles. (e) does not represent the same tangle as (a), since the labelling is different.
Definition 1.4. Rational tangles are 4-ended tangles without any closed components
obtained from the 4-ended tangle in figure 4a by adding twists to the top and to the right.
Remark 1.5. One might wonder why we have defined tangles the way we have. Alterna-
tively, we could have simply allowed only those isotopies that fix the whole boundary sphere.
However, such a definition is too rigid, since it differentiates between far too many tangles
that are essentially the same. In the other extreme, allowing just any isotopies would mean
that we do not distinguish between tangles that are related by some twists of the tangle
ends. As a consequence, all rational tangles would, for example, be the same, see figure 4.
The introduction of a fixed circle on the boundary sphere in definition 1.1 mediates between
the two extremes. It can be viewed as a parametrisation of the punctured sphere ∂B3r ∂T .
We explore this point of view in section I.4, in particular, see proposition 4.8. It will also
play a major role in chapters II and III.
Alexander polynomials of knots and links. Next, let us recall how the Alexander
polynomial of knots and links can be computed using Kauffman states and Alexander codes
following [Kau83]. Given a diagram of a 2-ended tangle (whose closure represents a knot or
link), a Kauffman state is an assignment of a marker • to one of the four regions at each
crossing such that each closed region is occupied by exactly one marker. One then applies
the Alexander codes to the Kauffman states, i. e. one labels the markers by the monomials
specified by the Alexander codes, as shown in figure 5b. To get the multivariate Alexander
polynomial, one just multiplies these labels, takes the sum over all Kauffman states and
finally multiplies everything by some normalisation factor.
When trying to apply this well-known algorithm to the general case of a 2n-ended tangle,
one encounters the following problem: There are, say, m crossings in the diagram, so by
an Euler characteristic argument, there are at least (m + n + 1) regions. (We have exactly
(m+n+1) regions iff all regions are simply connected. Otherwise, we have a split component
and so the Alexander polynomial should be zero.) Thus, there are at least (n + 1) regions
more than there are markers, but the number of open regions in a connected diagram is 2n.
This motivates the following two definitions.
Definition 1.6. A site of a 2n-ended tangle T is a choice of an (n− 1)-element subset
of the set of arcs S1 r im(T ). For connected tangle diagrams, this is equivalent to choosing
(n− 1) open regions. The set of all sites of a tangle T is denoted by S = S(T ).
12 I. ALEXANDER POLYNOMIALS FOR TANGLES
u
u−1
−u−1
u
(a) The Alexander code
for a positive crossing from
[Kau83, figure 33]. There is
a similar one for a negative
crossing. u is the colour of
the under-strand.
b
b
b
−t−1
t−1
t
(b) A Kauffman state for
a 2-ended tangle, coloured
by t. The closure is indi-
cated by the grey arc.
p
p
q
q
b
b
b
b
b
−p−1
p
p−1
p
q
(c) A Kauffman state for
a 4-ended tangle with two
components coloured by p
and q, respectively.
Figure 5. Applying Alexander codes to Kauffman states
Remark 1.7. Later, we will give a more geometric interpretation of sites, see proposi-
tion 4.8. In the categorified invariants of chapters II and III, sites correspond to idempotents
of the glueing algebra.
Definition 1.8. Let T be a (connected) diagram of an oriented 2n-ended tangle. A
generalised Kauffman state of T is an assignment of a marker to one of the four regions
at each crossing such that each closed region is occupied by exactly one marker, with the
additional condition that there be at most one marker in each open region. Furthermore,
• denote the set of all generalised Kauffman states of T by K = K(T ).
• For x ∈ K, let s(x) be the set of open regions that are occupied by a marker of x.
(Note that s(x) has (n− 1) elements, so s(x) ∈ S.)
• For each site s ∈ S, let K(T, s) := {x ∈ K(T )|s(x) = s}.
• For x ∈ K, let l(x) be the labelling of the markers of x according to the Alexander
codes in figure 6. Moreover, let c(x) be the product of the labels l(x).
Then for each site s ∈ S, let
∇ˆsT :=
∑
x∈K(T,s)
c(x).
Furthermore, let ∇sT denote the function ∇ˆsT evaluated at h = −1. We call ∇sT the Alex-
ander polynomial of T at the site s.
Remark 1.9. The variable h stands for “homological grading”. In chapter II, we will
generalise the hat version of knot and link Floer homology to tangles. The generators of
these homology groups will correspond to the generalised Kauffman states above.
Observation 1.10. In the Alexander codes of figure 6, the exponents of u in the two
regions left of an under-strand are −12 , and +12 in the regions on its right. For over-strands,
it is the other way round. This Alexander code has the advantage over the one in figure
5a that we do not need to multiply ∇sT by a normalisation factor to turn it into a tangle
invariant.
Theorem 1.11. For two oriented tangle diagrams T1 and T2 representing the same tangle
and s ∈ S(T1) = S(T2), we have ∇sT1 = ∇sT2. So ∇sT is a tangle invariant.
I.1. THE TANGLE INVARIANT ∇ST 13
o
1
2u
1
2
o
1
2u−
1
2
h−1o−
1
2u−
1
2
o−
1
2u
1
2
(a) A positive crossing
o−
1
2u−
1
2
o
1
2u−
1
2
ho
1
2u
1
2
o−
1
2u
1
2
(b) A negative crossing
Figure 6. The Alexander codes for definition 1.8. The variable o is the colour of
the over-strand and u the colour of the under-strand.
Proof. By lemma 1.3, we just need to check that the polynomials are invariant under
the Reidemeister moves RM I–III. We can check this locally, so the proof becomes exactly
the same as for the usual knot and link case: We verify the theorem for the basic diagrams
that appear in the Reidemeister moves, for each site separately. We only do this for RM I
and II here; for RM III, we refer the reader to the Mathematica notebook [APT.nb] which
uses the package [APT.m] for calculating ∇sT for any connected tangle diagram T and site s;
see also appendix C for the corresponding manual.
RM I looks as follows:
RM I
The enclosed region on the right only has one crossing. Hence, the corresponding marker
has to sit in that region in every Kauffman state. For both orientations, the labelling of
this marker is 1, so we might as well remove this crossing. The same holds if we reverse the
crossing; we can either check this directly, or apply proposition 2.1.
For RM II, we only check one orientation; again, for the others, we can either check this
separately or simply apply proposition 2.5.
p q
p q
RM II
p q
p q
In the diagram on the right, there are exactly two Kauffman states that occupy the open
region on the left; they contribute p and hp, so after setting h = −1, they cancel. The same
is true for the open region on the right; the contribution there is q and hq. Finally, for each
of the open regions at the top and the bottom, there is exactly one Kauffman state and it
contributes 1. 
We have chosen the letter ∇ for a reason:
Theorem 1.12. Let T be a diagram of an oriented 2-ended tangle representing a link L.
Note that in this case there is only one site of T , namely the empty set ∅. Let the colour of
the open component be c. Then the Conway potential function ∇L is equal to
1
c− c−1∇
∅
T .
14 I. ALEXANDER POLYNOMIALS FOR TANGLES
Proof. Verify that ∇ := 1
c−c−1∇∅T satisfies the axioms in [Jia14], see [APT.nb]. 
Remark 1.13. Recall that the Conway potential function of an n-component oriented
link L is a rational function ∇L(t1, . . . , tn) which is related to the multivariate Alexander
polynomial ∆L in the following way: (see for example [Har83] or [Jia14])
∇L(t1, . . . , tn) =

∆L(t
2
1)
t1 − t−11
, if n = 1;
∆L(t
2
1, . . . , t
2
n), if n > 1.
Hence, using the notation of the theorem above
∇∅T (t1, . . . , tn) =
∆L(t21), if n = 1;(c− c−1)∆L(t21, . . . , t2n), if n > 1.
We also note that ∇∅T of a 2-ended tangle T , multiplied by a factor of (c − c−1) for each
closed component of T , is equal to the Euler characteristic of Ozsváth and Szabó’s link Floer
homology from [OS05].
We collect some properties of the Conway potential function in the following theorem.
Theorem 1.14. [Har83, propositions 5.6, 5.5, 5.7 and 5.3] The Conway potential function
of an oriented link L satisfies the following properties:
(i) If m(L) denotes the mirror image of L, then
∇m(L)(t1, . . . , tr) = (−1)r−1 · ∇L(t1, . . . , tr).
(ii) ∇L(t1, . . . , tr) = (−1)r · ∇L(t−11 , . . . , t−1r ).
(iii) If r(L, t1) is obtained from L by reversing the orientation of the first strand, then
∇r(L,t1)(t1, . . . , tr) = −∇L(t−11 , t2, . . . , tr).
(iv) If L1, . . . , Lr denote the link components of L, then ∇L(1, t2, . . . , tr) is equal to
 (tlk(L1,L2)2 · · · tlk(L1,Lr)r − t− lk(L1,L2)2 · · · t− lk(L1,Lr)r )∇LrL1(t2, . . . , tr). 
Often, we want to glue tangle diagrams together along some parts of their boundary to
obtain a new tangle diagram. We can easily compute the polynomial invariant of the new
tangle from the invariants of the two glued components. (In fact, we have implicitly used
the fact that ∇sT behaves well under glueing in the proof of theorem 1.11 already.) The
following result generalises the connected sum formula for knots and links.
Proposition 1.15 (splitting/glueing formula). Let T1 and T2 be two oriented tangles
obtained by splitting an oriented tangle diagram T along some arc that does not meet any
crossings, for example like so:
T = T1 T2
I.2. BASIC PROPERTIES OF ∇ST 15
Let t1,1, . . . , t1,r1 , t2,1, . . . , t2,r2 and t1, . . . , tr be the colours of T1, T2 and T , respectively.
Glueing T1 and T2 back together induces an identification of these colours which gives rise
to two homomorphisms
ιi : Z[t
±1/2
i,1 , . . . , t
±1/2
i,ri
]→ Z[t±1/21 , . . . , t±1/2r ], i = 1, 2.
Then
∇ˆsT =
∑
s1∩s2=∅
(s1∪s2)∩O(T )=s
ι1(∇ˆs1T1) · ι2(∇ˆs2T2),
where O(T ) denotes the set of open regions in the diagram T .
Proof. This follows immediately from definition 1.8, noting that the sum is over exactly
those marker assignments that define generalised Kauffman states for T . 
Before moving on to the next section to study some properties of our tangle invariant,
we state a generalisation of Kauffman’s clock theorem [Kau83, theorem 2.5] to tangles. For
this we recall the following definition from [Kau83, figure 5]. (Note that the theorem is false
if we use the stronger definition given in the introduction of the same monograph.)
Definition 1.16. Suppose in a tangle diagram, there are two crossings which have two
regions in common. Suppose further that in a Kauffman state x, the markers of the two
crossings lie in these two regions. Then there also exists a Kauffman state y obtained from
x by moving each of the markers of the two crossings in exactly the opposite region. We say
that we go from x to y by a transposition move. A transposition move is called clockwise
if the markers move clockwise around each crossing, as illustrated below:
b
b
x −→ y
b
b
The reverse is called a anticlockwise move. In appendix B, we prove the following result.
Theorem 1.17 (generalised clock theorem). Let T be a (not necessarily connected) tangle
diagram and s ∈ S(T ). Then K(T, s) is a finite lattice under the relation
x > y ⇔ ∃ sequence of clockwise transposition move from x to y.
I.2. Basic properties of ∇sT
In this section, we collect and prove some properties of the tangle invariants ∇sT ,
guided by theorem 1.14. We will see that with the exception of the symmetry prop-
erty (ii), the results generalise to the tangle case. Our generalisation of property (iv)
leaves room for improvement because it only applies to closed components. For open com-
ponents, some better understanding of the relations between sites would probably be helpful.
The first proposition below corresponds to part (i) combined with (ii) of theorem 1.14.
Proposition 2.1. Let T be an oriented tangle and m(T ) its mirror image. Then for all
s ∈ S(T ),
∇ˆsm(T )(t1, . . . , tr, h) = ∇ˆsT (t−11 , . . . , t−1r , h−1).
16 I. ALEXANDER POLYNOMIALS FOR TANGLES
Proof. Observe that the two Alexander codes in figure 6 are mirror images of one
another after taking the reciprocals of all variables. 
Definition 2.2. We define the linking number lkT (p, q) for two components p and q
of a tangle T to be
lkT (p, q) :=
1
2(#{positive crossings of p and q} −#{negative crossings of p and q}).
For a tangle with a component tj , we also define
lkT (tj) :=
∑
lkT (ti, tj),
where the sum is over all components ti 6= tj . We sometimes omit the subscript when there
is no risk of ambiguity.
Remark 2.3. Note that for two-component links, lk(p, q) coincides with the usual linking
number. Also, linking numbers are invariants of tangles.
Lemma 2.4. Given a tangle diagram T , the powers of any colour in two Kauffman states
of the same site s ∈ S(T ) differ by a multiple of 2. Furthermore, the exponent of a colour t
in ∇ˆsT is an integer iff lk(t) is an integer.
Proof. By the generalised clock theorem (theorem 1.17), any two Kauffman states with
site s are connected by a sequence of transposition moves. It is easy to see that two states
connected by a single transposition move have either the same Alexander grading or the
exponents of one colour (the one corresponding to the horizontal strand) changes by ±2.
The second statement follows directly from the definition of the linking number. 
The next proposition corresponds to part (iii) of theorem 1.14.
Proposition 2.5. Let T be an oriented r-component tangle. If r(T, t1) denotes the
same tangle T with the orientation of the first strand reversed, then for all sites s ∈ S(T ),
we have
∇ˆsr(T,t1)(t1, . . . , tr) = hlkT (t1)∇ˆsT (h−1t−11 , t2, . . . , tr).
Proof. This is easily seen by considering crossings separately: Modulo sign, the state-
ment follows from observation 1.10. For the correct sign, note that after substituting h−1t−11
into the Alexander code of a positive (negative) crossing involving t1 and some different
colour, we obtain the Alexander code of the crossing with the orientation of the t1-strand
reversed multiplied by h−
1
2 (respectively h
1
2 ). For crossings involving only t1, no additional
factor is necessary, and for crossings not involving t1 at all, there is nothing to show. 
Note that one has to be careful when setting h = −1 in the proposition above, because
the ∇ˆsT are Laurent polynomials with half-integer powers. The same applies to corollary 2.6
below. In corollary 2.13, we give a formula that is more convenient when working with ∇sT
instead of ∇ˆsT .
Corollary 2.6. Let T be an oriented r-component tangle. If r(T ) denotes the same
tangle T with the orientation of all strands reversed, then for all sites s ∈ S(T ), we have
∇ˆsr(T )(t1, . . . , tr) = ∇ˆsT (h−1t−11 , . . . , h−1t−1r ).
I.2. BASIC PROPERTIES OF ∇ST 17
Proof. We successively reverse the orientation of all strands, noting that each term
lkT (ti, tj) appears twice in the exponent of h, but with different signs, because the second
time it appears, the orientation of one strand has been reversed. 
Corollary 2.7. Let r(·) denote the function which substitutes −t−1 for each colour t.
Then, for an oriented link L, we have the symmetry relation
∇L = ∇r(L) = r(∇L).
Proof. The first equality is theorem 1.14 (ii) and (iii), the second follows from theo-
rem 1.12 and the corollary above with h = −1. 
Lemma 2.8 (one-colour skein relation). Let T+, T− and T◦ denote the tangles ,
and respectively. Then for all sites s,
∇sT+(t, t)−∇sT−(t, t) = (t− t−1) · ∇sT◦(t, t).
Thus, the single-variate polynomial tangle invariant ∇sT (t, . . . , t) satisfies the same skein
relation as the Alexander polynomial.
Proof. Straightforward. 
Corollary 2.9. Let T be a 2-ended tangle representing a knot. Then ∇sT (±1) = 1.
Proof. Let T ′ be the diagram obtained from T by changing some crossings such that
T ′ represents the unknot. Then, by lemma 2.8, ∇sT (±1) = ∇sT ′(±1), and ∇sT ′(t) ≡ 1. 
The next proposition corresponds to part (iv) of theorem 1.14.
Proposition 2.10. Let T be a tangle with a closed component K1. Then for all sites
s ∈ S, ∇sT (± 1, t2, . . . , tr) is equal to
(± 1)lk(t1)+1(tlk(t1,t2)2 · · · tlk(t1,tr)r − t− lk(t1,t2)2 · · · t− lk(t1,tr)r ) · ∇sTrK1(t2, . . . , tr).
Proof. First we apply lemma 2.8 to t1-t1-crossings in T . Setting t1 = ± 1 gives
∇sT (± 1, t2, . . . , tr) = ∇sT ′(± 1, t2, . . . , tr),
where T ′ denotes the tangle obtained by swapping over- and under-strands at some t1-
t1-crossing. We can therefore assume without loss of generality that the t1-component is
the unknot, in particular that there are no t1-t1-crossings and near the t1-component, the
diagram looks as follows:
T±
T±
T±
T±
T±
where T+ =
t1 ti
and T− =
t1 ti
for i 6= 1.
18 I. ALEXANDER POLYNOMIALS FOR TANGLES
The following table gives the values of ∇sT± , where l, b, r, t denote the regions on the left,
bottom, right and top of the diagrams T±, respectively.
s T+ T−
l (ti − t−1i ) −(ti − t−1i )
b t−11 t
−1
i t
−1
1 ti
r (t1 − t−11 ) −(t1 − t−11 )
t t1ti t1t
−1
i
so after setting t1 = ±1:
s T+ T−
l (ti − t−1i ) −(ti − t−1i )
b ± t−1i ± ti
r 0 0
t ± ti ± t−1i
Let the number of rectangular boxes T± be denoted by n. The fact that ∇rT±(1, ti) = 0
means that if we consider the picture above as a 2n-ended tangle T ′′, there are at most n
sites such that the polynomial invariant becomes non-zero after substituting t1 = ±1. Each
of these sites is specified by the region which is not occupied by a marker and which is not
a right region of any box T±. Fix such a region u. Then each Kauffman state of T ′′ is
determined by a box and a marker in the left region of this box. For each box, there are two
such Kauffman states, namely those that contribute to ∇l of that box.
We introduce the following notation: We number the boxes in anticlockwise direction, start-
ing from the fixed unoccupied region u. Let αi±j denote the number of T±s involving the
ith strand in anticlockwise direction from u to the jth box Tj . Similarly, let βi±j denote
the number of T±s involving the ith strand in clockwise direction from u to Tj . Let ij 6= 1
denote the index of the strand involved in Tj . The contribution of the boxes T± 6= Tj to the
labelling of the two Kauffman states corresponding to Tj is
αi+j → ± ti,
αi−j → ± t−1i ,
βi+j → ± t−1i ,
βi−j → ± ti,
so that the contribution of the two Kauffman states to ∇T ′′ of the site determined by u is
±(tij − t−1ij ) ·
∏
i 6=1
(±ti)(α
i+
j +β
i−
j )−(αi−j +βi+j ).
Note that by definition
lk(t1, ti) =

αi+j − αi−j + βi+j − βi−j if i 6= 1, ij ,
αi+j − αi−j + βi+j − βi−j + 1 if i = ij and Tj = T+,
αi+j − αi−j + βi+j − βi−j − 1 if i = ij and Tj = T−.
So the expression above becomes
±(tij − t−1ij ) · (± tij )
lk(t1,tij )−2(β
ij+
j −β
ij−
j )∓ 1 ·
∏
i 6=1,ij
(± ti)lk(t1,ti)−2(β
i+
j −βi−j )
= ±
(
(± tij )lk(t1,tij )−2(β
ij+
j −β
ij−
j ) − (± tij )lk(t1,tij )−2(β
ij+
j −β
ij−
j ± 1)
)
·
∏
i 6=1,ij
(± ti)lk(t1,ti)−2(β
i+
j −βi−j ).
I.2. BASIC PROPERTIES OF ∇ST 19
Now, consider two consecutive boxes, say the jth and the (j + 1)st box. Then
β
ij+
j − βij−j = βij+j+1 − βij−j+1± 1.
and for all other colours, the exponents remain the same. So we see that most terms cancel
and the only surviving ones are the second one of the first box and the first one of the last
box. But
βi+1 − βi−1 =
lk(t1, ti) if i 6= 1, i1,lk(t1, ti1)∓ 1 if i = i1,
and
 βi+n = βi−n = 0 for all i 6= 1. 
Proposition 2.11. Given a tangle T , we can compute the powers modulo 2 appearing
in the labellings of Kauffman states as follows. If t is the colour of a closed component in T ,
its exponents are equal to lk(t) + 1 mod 2. If it is the colour of an open component, its
exponents are equal to
lk(t) + 1 + 12(si − so + ro − ru − 1) ≡ lk(t) + 1 + si + ru mod 2,
where, if we follow the outgoing t-strand in anticlockwise direction along the boundary of
the disc to the other end, si is the number of ingoing strands, so is the number of outgoing
strands, ro is the number of occupied regions and ru is the number of unoccupied regions that
we meet along the way.
Remark 2.12. It is easy to see that the formula is symmetric in the sense that it does
not matter if we go clock- or anticlockwise. Nonetheless, it would be nice to find a more
natural interpretation of this formula. Again, a better understanding of the relationship
between sites would probably be useful for this.
Corollary 2.13 (reversing orientations revisited). Let T be an oriented r-component
tangle. If r(T, t1) denotes the same tangle T with the orientation of the first component K1
reversed, then for all sites s ∈ S(T ), we have
∇sr(T,t1)(t1, . . . , tr) =
−∇sT (t−11 , t2, . . . , tr) K1 closed,(−1)2 lkT (t1)+1+si+ru∇sT (t−11 , t2, . . . , tr) K1 open,
where we use the same notation as in the previous proposition. Similarly, if r(T ) denotes
the same tangle T with the orientation of all strands reversed, then for all sites s ∈ S(T ),
we have
∇sr(T )(t1, . . . , tr) = (−1)r+c+
∑
si+
∑
ru∇sT (t−11 , . . . , t−1r ),
where c ≡∑ lk(ti) mod 2 is the number of interlinked pairs of endpoints of the same colour
on the boundary of the disc.
Proof. The first part follows directly from proposition 2.5 and 2.11. For the second
part, we apply proposition 2.11 to corollary 2.6. 
Before we come to the proof of proposition 2.11, we do some preparation first.
Lemma 2.14. Let T be a tangle with no closed components. Then there exists a site
s ∈ S(T ) such that ∇sT 6≡ 0.
20 I. ALEXANDER POLYNOMIALS FOR TANGLES
Proof. We extend the diagram T to one of a knot by successively connecting two ends
of strands with different orientations and colours until only two ends are left. (Note that
this process might introduce some more crossings.) Then ∇s of this knot will be a linear
combination of the ∇sT . Since it is also non-zero (say, by corollary 2.9), ∇sT cannot be
identically zero for all s. 
Lemma 2.15. Let T be a (connected) tangle diagram (with at least one crossing). Then
there exists a site s such that K(T, s) is non-empty.
Proof. As in the previous proof, we close all but two strands of the diagram in some
fashion. This new diagram represents a link and, by [Kau83, lemma 2.2, theorem 2.4], it has
a Kauffman state. Hence its restriction to the original tangle diagram is also a Kauffman
state for some site s. 
Lemma 2.16. Given a tangle T , let x and x′ be two Kauffman states whose sites s
and s′ differ in exactly two adjacent open regions, separated by a t-coloured strand. Then the
exponent of t in the labelling of x and x′ differs by 1 modulo 2. Exponents of other colours
agree modulo 2.
Proof. As in the proof of lemma 2.4, we would like to have some simple move that we
can perform to get from one site to another and that affects the labels in a predictable way
and then appeal to some connectedness result. Here, the basic move is the following, we call
it the boundary move:
b
∂D2
←→
b
∂D2
Consider two Kauffman states which are related by a single boundary move. The labellings
of these two Kauffman states differ by the factor t±1.
However, such a boundary move might not always be possible. To fix this, we first modify
the diagram slightly, namely we perform a Reidemeister II move as follows:
b
∂D2
−→
b
b
b
∂D2
The labelling of the new Kauffman state corresponding to x as shown in the right picture
above is the same as the labelling of x itself. The same is true for the labelling of x′. (The
additional two markers are then on the “top” of the new crossings.) We can now perform
a boundary move as indicated by the arrow in the picture on the right. By lemma 2.4, the
powers of each colour in the labellings of this Kauffman state and x′ agree modulo 2. 
I.2. BASIC PROPERTIES OF ∇ST 21
∂D2
b
t−
1
2
occupied
t
negative crossing
∂D2
b
t−
1
2
occupied
t
positive crossing
∂D2
b
t
1
2
unocc.
t
negative crossing
∂D2
b
t
1
2
unocc.
t
positive crossing
Figure 7. Four cases distinguished in the proof of proposition 2.11
Lemma 2.17. For any (not necessarily connected) tangle diagram T , we can find an
equivalent diagram T ′ such that for any site s ∈ S(T ) = S(T ′), T ′ has a Kauffman state.
Furthermore, the labelling of any Kauffman state of T agrees with the labelling of any Kauff-
man state in T ′ belonging to the same site, considering any exponents modulo 2.
Proof. If there is no Kauffman state in T , we can modify the diagram such that the
hypotheses of lemma 2.15 are satisfied. So we may assume without loss of generality that
there exists a Kauffman state in T for some site. We now repeatedly apply the same method
as in the proof above, noting that a Reidemeister II move merely enlarges the set of Kauffman
states and the labellings of corresponding Kauffman states agree. 
Proof of proposition 2.11. Let s be the site of a Kauffman state x. We consider
closed components first. Suppose that all linking numbers are non-zero. Then the claim
follows from proposition 2.10 and lemma 2.4 if ∇s of the tangle T with all closed components
removed is not identically zero. By lemma 2.14, we can find at least one site s′ of this tangle,
such that for any Kauffman state in K(T, s′), the claim is true. By lemma 2.17, we may
assume that there is a Kauffman state for each site. Then by lemma 2.16, we know that in
particular the exponents of the colours of closed components agree for all sites.
The general case, where linking numbers may be zero, can be reduced to the first case. For
this, we modify the diagram as follows: Consider two strands whose mutual linking number
is zero. If they meet at a crossing, we can reverse the over- and under-strands at one such
crossing. This changes both the linking number and the exponents of both colours by ±1.
If two components have no crossing in common, we perform some Reidemeister II moves to
pull one strand across the other to get one. Note that this does not affect the labelling of the
Kauffman states (as in the previous proof). This finishes the proof for closed components.
For open components, the idea is to add twists at the outgoing strand until it bounds the
same open region as the incoming strand, so that for some sites, we can close this component
and apply the result above. We use the convention that the t-strand always goes over its left
neighbour. We can then distinguish the cases shown in figure 7, depending on the orientation
of the other strand involved and on whether the open region between these two strands is
occupied by a marker or not.
Now, suppose the open region on the left of the incoming t-strand is occupied. Then we
can close the t-component and apply the result above. The exponent of t in the original
diagram is equal to lknew(t)+1+ 12((ro−1)−ru), where lknew(t) denotes the linking number
22 I. ALEXANDER POLYNOMIALS FOR TANGLES
of the closed t-component in the new diagram. Suppose the open region on the left of the
incoming t-strand is unoccupied, so we cannot close the t-component. One way to solve
this problem is by “pushing” a marker of an open region into this region, using lemmas 2.17
and 2.16. However, this does not work for 2-ended tangles where we do not have any
markers to “push around”. Although one can, of course, look at this case separately, we give
an alternative argument that works in general: We still close the t-component, but instead
of glueing a single strand to the diagram, we attach a diagram that consists of two parallel
strands that are twisted by a Reidemeister II move. By considering ∇ for the attached
two-crossing diagram, we see that the exponent of t in the original Kauffman state is given
by lknew(t) + 1 + 12(ro − (ru − 1)) + 1 mod 2, i.e. the same formula as in the first case.
So it remains to calculate lknew(t). We have lknew(t) = lkT (t) + 12(si − so). Substituting
this into the formula above gives the first formula in the statement of the proposition. The
equivalence to the second one is seen by substituting the obvious identity si+so = ro+ru−1
into the first formula (and adding 2ru). 
I.3. 4-ended tangles and mutation invariance
Proposition 3.1. Let T be a 4-ended tangle. Then the endpoints of the same strands
are either neighbours on ∂D2 (case I) or not (case II). We consider the following orientations
for these two cases:
c
d
a
b
p
p
q
q
case I
c
d
a
b
p
p
q
q
case II
Then in both cases,
(b-d) ∇bT = ∇dr(T ).
Furthermore, in case I
(p− p−1) · ∇aT = (q − q−1) · ∇cT ,(I a-c)
∇cT = ∇cr(T ) and(I c-c)
(pq − p−1q−1) · ∇cT = (p− p−1)
(
∇dT −∇dr(T )
)
(I c-d)
and in case II
∇aT = ∇cr(T ),(II a-c)
(pq − p−1q−1) · ∇cT = (p− p−1) · ∇dT − (q − q−1) · ∇dr(T ) and(II c-d)
(p−1q − pq−1) · ∇dT = (q − q−1) · ∇aT − (p− p−1) · ∇ar(T ).(II d-a)
For orientations different from the above, we get similar relations, which we can easily com-
pute using proposition 2.5 and 2.11.
I.3. 4-ENDED TANGLES AND MUTATION INVARIANCE 23
a
b
c
d
Figure 8. The trefoil knot with an additional strand, illustrating remark 3.3
Corollary 3.2. Let T be a 4-ended tangle. We distinguish between the following two
cases:
t
t
t
t
c
d
a
b
case ii
t
t
t
t
c
d
a
b
case i
We write ∇a = ∇aT (t, t, t1, . . . , tr−2), ∇ar = ∇ar(T )(t, t, t1, . . . , tr−2) and similarly for b, c
and d. Then in both cases, we have ∇c = ∇a = ∇ar , ∇b = ∇dr and (t+ t−1)∇c = ∇d −∇dr ,
so
t∇a +∇b + t−1∇c = ∇d.
In case ii, we additionally get ∇b = ∇d.
Proof. The first identity is seen by closing a or c. The second follows from the properties
of the Conway potential function, theorem 1.14(ii) and (iii). The third and fourth follow
from proposition 3.1. The last relation is seen again by closing b or d. 
Remark 3.3. Proposition 3.1 tells us that there is basically only one piece of information
in the polynomial Alexander invariant of 4-ended tangles. One might ask whether all four
invariants in case I contain the same information like in case II. However, this is not the
case: For any knot K, we can consider the two-component tangle diagram obtained from
K by cutting it once to get one strand of the tangle, adding an unknotted strand to it and
performing a Reidemeister II move to get rid of any multiple regions on the boundary, like
in figure 8 for K = trefoil knot. Then both a and c are zero because closing either of these
regions results in a link diagram with two unlinked components. However, for matching
orientations of the strands, b and d both give the Alexander polynomial ∆K of the knot K.
Proof. Relation (I a-c) follows from theorem 1.12 and the fact that the two diagrams
obtained by closing either the p-component or the q-component both represent the same
link. Next, using the notation from corollary 2.7 and the corollary itself, we obtain
r
( ∇cT
p− p−1
)
=
∇cT
p− p−1 .
so we immediately get (I c-c).
24 I. ALEXANDER POLYNOMIALS FOR TANGLES
Next, we compute the invariants for the following tangle U :
∇AU = (q − q−1)
∇BU = p−1q−1
∇CU = (p− p−1)
∇DU = pq
D
B
A C
p q
p q
We observe that the diagrams obtained by glueing the above diagram either to the top or to
the bottom of T and closing the q-component represent the same link. In the first case, the
corresponding polynomial is given by∇CU ·∇dT +∇BU ·∇cT and the second, by∇CU ·∇bT +∇DU ·∇cT .
Again, corollary 2.7 implies
(1) r
(∇CU ∇dT +∇BU ∇cT
p− p−1
)
=
∇CU ∇dT +∇BU ∇cT
p− p−1 =
∇CU ∇bT +∇DU ∇cT
p− p−1 ,
After some simplification, we get
r(∇CU ) · ∇dr(T ) + r(∇BU ) · ∇cr(T ) = ∇CU · ∇bT +∇DU · ∇cT
Finally, using r(∇BU ) = ∇DU and r(∇CU ) = ∇CU , we obtain the desired identities (b-d) for case
I. For (I c-d), we just use the first half of equations (1) and, after simplification, we get
r(∇CU ) · ∇dr(T ) + r(∇BU ) · ∇cr(T ) = ∇CU · ∇dT +∇BU · ∇cT .
Substituting ∇cr(T ), r(∇BU ) and r(∇CU ) gives the desired identity.
By a similar method, we can derive the relations for case II. We first show the second relation.
For this, we apply the corresponding statement from case I to the diagram obtained by
glueing the following positive twist to the bottom of T :
p
1
2 q
1
2
p
1
2 q−
1
2
−p− 12 q− 12
p−
1
2 q
1
2
Then, we have
−p− 12 q− 12∇bT = r
(
p
1
2 q
1
2∇dT
)
,
which is (b-d). (II c-d) follows similarly; we have
(pq − p−1q−1) ·
(
p
1
2 q
1
2∇cT + p−
1
2 q
1
2∇bT
)
= (p− p−1)
(
p
1
2 q
1
2∇dT − r
(
p
1
2 q
1
2∇dT
))
.
Substituting ∇bT yields
(pq − p−1q−1)∇cT = (p− p−1)∇dT + (q−1 − p−2q−1) · ∇dr(T ) − (q − p−2q−1) · ∇dr(T )
= (p− p−1)∇dT − (q − q−1)∇dr(T ).
For (II a-c), we compare the two diagrams obtained by glueing a single positive twist either
to the top or to the bottom of T and closing the q-strand.
I.3. 4-ENDED TANGLES AND MUTATION INVARIANCE 25
We obtain
r
(
−p− 12 q− 12∇aT + p
1
2 q−
1
2∇dT
p− p−1
)
=
p
1
2 q
1
2∇cT + p−
1
2 q
1
2∇bT
p− p−1 ,
This simplifies to
p
1
2 q
1
2∇ar(T ) + p−
1
2 q
1
2∇dr(T ) = p
1
2 q
1
2∇cT + p−
1
2 q
1
2∇bT
Using (b-d), the result follows. Finally, relation (II d-a) is obtained from the previous one
using proposition 2.5 and 2.11. We reverse the p-strand, so we get
(pq−1 − p−1q) · ∇cr(T,p) = (p− p−1) · ∇dr(T,p) − (q − q−1) · ∇dr(r(T ),p).
We observe that r(T, p) also belongs to case II; however, to get the same configuration as
in the proposition, we have to rotate it by 90◦ anticlockwise and switch p and q. Doing this
for the identity above gives us the required result. 
Definition 3.4. Let T be a tangle and T◦ a 4-ended tangle obtained by intersecting T
with a closed 3-ball B3. We may assume that all four tangle ends of T◦ lie equally spaced
on a great circle on ∂B3. Let T ′ be the tangle obtained from T by rotation of T◦ by pi about
one of the three axes that switch pairs of endpoints of T◦ as shown in figure 9. We say T ′ is
obtained from T by mutation or T ′ is a mutant of T . T◦ is called the mutating tangle.
If T is oriented, we choose an orientation of T ′ that agrees with the one for T outside of
B3. If this means that we need to reverse the orientation of the two open components of T◦
then we also reverse the orientation of all other components of T◦ in T ′; otherwise we do not
change any orientation.
Remark 3.5. The definition above is equivalent to the one given in the introduction for
links (definition 0.2). This can be easily seen by twisting the ends of a mutating tangle, see
remark III.4.5.
Theorem 3.6. Let T be an oriented tangle and T ′ a mutant of T . Suppose the colours
of the two open strands of the mutating tangle agree. Then for all sites s ∈ S(T ) = S(T ′),
∇sT = ∇sT ′ .
Corollary 3.7. The multivariate Alexander polynomial is mutation invariant, provided
that the open strands of the mutating tangle have the same colour. 
x-axis
y-axis
z-axis
T◦
A
B
C
D
bbb
Figure 9. The three mutation axes
26 I. ALEXANDER POLYNOMIALS FOR TANGLES
Remark 3.8. It is quite easy to find a counterexample for a potential symmetry relation
∇sr(T,t) = ±∇sT , where t is a closed strand. For example, let
T = a
b
c
dp
p
r
Then ∇bT = p2r − (r − r−1) − p−2r−1. So in general, the theorem and its corollary above
become false if we do not reverse the orientation of all strands in the tangle T◦ when the
orientation of the open strands needs to be reversed for a mutation.
Proof of theorem 3.6. We consider the same two cases as in corollary 3.2, and also
use its notation. Denote by A, B, C and D the Alexander polynomials of the corresponding
counterparts of the sites a, b, c and d in T r T◦, such that
∇sT = A∇a +B∇b + C∇c +D∇d.
If we rotate about the x-axis, we have to reverse orientations in both cases of corollary 3.2,
so
∇sT ′ = A∇ar +B∇dr + C∇cr +D∇br = A∇a +B∇b + C∇c +D∇d = ∇sT .
Next, let us consider rotations about the y-axis. In case i, we do not need to reverse
orientations. We have
∇sT ′ = A∇c +B∇b + C∇a +D∇d = A∇a +B∇b + C∇c +D∇d = ∇sT .
In case ii, we need to reverse orientations:
∇sT ′ = A∇cr +B∇br + C∇ar +D∇dr = A∇a +B∇b + C∇c +D∇d = ∇sT
Finally, note that rotation about the z-axis is the same as rotation about both the x- and
the y-axis (in any order), so we are done. 
I.4. Geometric interpretation of ∇sT
Recall that the classical Alexander polynomial of a knot or link L can be defined as
follows: Let X˜L denote the maximal Abelian cover of the link complement XL = S3 r L.
We have an action of H1(XL) on X˜L. Thus, we can regard H1(X˜L) as a module over the
group ring of H1(XL). Then, the Alexander polynomial is the determinant of any square
presentation matrix of H1(X˜L).
In this section, we will give a similar geometric interpretation of our polynomial tangle
invariants. But first, we do some basic calculations.
Lemma 4.1. Let T be a tangle in B3 with n open and m closed components. Then
H1(B
3 r T ) ∼= Zn+m is freely generated by the meridians of the tangle components and
H2(B
3 r T ) ∼= Zm is freely generated by the boundaries of tubular neighbourhoods of the
closed tangle components.
I.4. GEOMETRIC INTERPRETATION OF ∇ST 27
a
b
c
d
(a) A positive crossing
a
b
c
d
(b) A negative crossing
Figure 10. Attaching 2-handles at crossings
Proof. The Mayer-Vietoris sequence for the decomposition B3 = (B3rT )∪ν(T ), ν(T )
being a tubular neighbourhood of T , gives isomorphisms
H∗((qnS1)q (qmT 2)) ∼= H∗(∂ν(T ))→ H∗(B3 r T )⊕H∗(ν(T )) for ∗ > 0.
Note that ν(T ) ' (qnB3) q (qmS1 × D2) and the longitude of any torus T 2 goes to the
generator of the corresponding H1(S1 ×D2). 
Next, we explicitly calculate the cellular chain complex of the tangle complement XT =
B3 r T from a fixed tangle diagram T by considering the following handle decomposition:
We start with two 0-handles, one sitting below and the other above the diagram. For each
region, take a 1-handle from one 0-handle to the other. To simplify arguments below, we
orient these 1-handles as follows: Choose a chequerboard colouring of the diagram and orient
the 1-handles corresponding to one colour in one direction and the others in the opposite
direction. Finally, for each crossing, attach a 2-handle to the 1-handlebody as illustrated in
figure 10, where a, b, c and d denote 1-handles. Up to an overall sign, the attaching map
is then given by a + b + c + d in both cases. This gives rise to the following cellular chain
complex of XT :
(2) Zc Zc+n+1 Z2,A
where c is the number of crossings, c+ n+ 1 the number of regions in the diagram and A is
the (c+n+1)×c matrix determined by the rules above. (Here, we assume that the diagram
is connected and has at least one crossing.) Let R = ZH1(XT ), the group ring of H1(XT ).
Then the cellular chain complex of the maximal Abelian cover is
(3) Rc Rc+n+1 R2.A˜
The attaching maps of the 1-handles, which determine the matrix A˜, are given by a +
o−1b + o−1c + d for positive crossings and a + b + o−1c + o−1d for negative crossings up to
multiplication by a unit in R, where o is the homology class of the meridian of the over-
strand. Also note that we have used the right-hand rule to determine the orientation of
these meridians. Each site s of a tangle diagram gives rise to a subhandlebody of XT which
we can regard as a subspace Bs of ∂B3r∂T : We associate with s the subhandlebody of XT
consisting of the two 0-handles and all 1-handles corresponding to those regions not in s,
i. e. unoccupied open regions. The cellular chain complex of Bs is
Zn+1 Z2,
28 I. ALEXANDER POLYNOMIALS FOR TANGLES
T
b
b
b
ec
mc
j
e+
e−
Figure 11. The cell decomposition near the meridian Mc used in observation 4.3
which we can consider as a subcomplex of (2). Similarly, we can consider the preimage B˜s of
Bs in X˜T and regard its cellular chain complex as a subcomplex of (3). Then the quotients
of these chain complexes calculate H∗(XT , Bs) and H∗(X˜T , B˜s), respectively, and they are
given by
Zc ZcAs and Rc Rc,A˜s
where As and A˜s denote the matrices obtained from A and A˜ by deleting those rows cor-
responding to s, respectively. Note that H1(X˜T , B˜s), considered as an R-module, is an
invariant of (B˜s ⊂ X˜T ), and A˜s is just a presentation matrix of H1(X˜T , B˜s).
Proposition 4.2. Let s be a site of a connected tangle diagram T . Let A˜s be a presen-
tation matrix of H1(X˜T , B˜s) as in the construction above. Then
det A˜s(t
2
1, . . . , t
2
r) =˙ ∇sT (t1, . . . , tr),
where =˙ denotes equality up to multiplication by a unit.
Observation 4.3. Let us verify this statement for knots and links. For a 2-ended tangle
T representing a knot or link L, define XL := B3 r T ∼= S3 r L. Hence the first homology
groups of the maximal Abelian covers are the same as H1(XL)-modules. Bs is homotopic to
the meridian of the open component, hence for knots, B˜s is homotopic to the real line, so
H1(X˜L, B˜s) = H1(X˜L). The argument for links is slightly more complicated. This is to be
expected because by remark 1.13, we should see an additional factor (c− 1), where c is the
colour of the single open strand.
Let us consider a slightly more general situation, where we have an arbitrary tangle T with a
closed component c and we want to compare H1(X˜T , B˜s) with H1(X˜T , B˜s q M˜c), where Mc
is a meridian of c. Given a diagram of T , consider the following handle-decomposition of XT :
Start with three 0-handles, one below the diagram on the boundary of the closed 3-ball (e−),
one above (e+) and one on Mc (ec). Then add a 1-handle for each open region, connecting
the two 0-handles above and below, a 1-handle mc along Mc and a 1-handle j joining ec to
e−, say. Finally, add a 2-handle along mj , j and the two 1-handles corresponding to the two
open regions on either side of Mc and additional 2-handles at the crossings as before. The
cell decomposition near the meridian Mc is illustrated in figure 11. Let R := H1(XT ). For
the maximal Abelian cover, we obtain the following chain complex:
Ra Ra+n ⊕Rmc ⊕Rj Re− ⊕Re+ ⊕Rec.A B
I.4. GEOMETRIC INTERPRETATION OF ∇ST 29
The matrix B looks follows: ∗ 0 1∗ 0 0
0 (c− 1) −1
 .
On the one hand, to get a presentation matrix ABsqMc for H1(X˜T , B˜sq M˜c), we can simply
take the quotient of this complex by
Rn+1 ⊕Rmc Re− ⊕Re+ ⊕Rec,
where Rn+1 ⊂ Ra+n is generated by those 1-handles corresponding to the site s; so we just
need to delete certain rows in A. On the other hand, if we quotient only by a subcomplex
of the previous one, namely
Rn+1 Re− ⊕Re+,
then the kernel of the right-hand map in the quotient complex
Ra Ra+n/Rn+1 ⊕Rmc ⊕Rj RecAq Bq
is Ra+n/Rn+1 ⊕ R(mc + (c − 1)j). By doing a simple change of basis that sends mc to
mc + (c − 1)j and fixes all other basis vectors, we see that the presentation matrix of
H1(X˜T , B˜s) is obtained from Aq by deleting the row for j. Now the fact that BqAq = 0
implies that the row for mc in Aq multiplied by (c−1) is equal to the row for j in Aq. Thus,
by linearity of the determinant with respect to rows, we obtain
det(ABsqMc) = (c− 1) det(ABs).
We will use this argument later in the proof of theorem II.1.2.
The proof of proposition 4.2 follows from basically the same argument as the corre-
sponding statement for the classical Alexander polynomial, see [Kau83, proposition 3.1].
The crucial point is that the signs in the definition of the determinant of a matrix and the
signs coming from h = −1 in the Alexander codes correspond to one another. In order to
see this, we make use of the generalised clock theorem again. We also need the following
lemma to match the Alexander codes – and again, the generalised clock theorem is the key.
Lemma 4.4. In definition 1.8, we can replace the Alexander codes from figure 6 by those
in figure 12 to obtain new polynomials ∇sT,new. Then for all tangle diagrams T and sites s
∇sT,new(t21, . . . , t2r) =˙ ∇sT (t1, . . . , tr).
o
1
2
o
1
2
−o− 12
o−
1
2
(a) A positive crossing
o−
1
2
o
1
2
−o 12
o−
1
2
(b) A negative crossing
Figure 12. The alternative Alexander codes from lemma 4.4. The variable o is the
colour of the over-strand.
30 I. ALEXANDER POLYNOMIALS FOR TANGLES
Proof. Using observation 1.10, we consider the effect of a transposition move on the
labelling of Kauffman states. Let the colour of the horizontal strand in figure 13 be t. Note
that any other colours are not affected by a transposition move. Now, if the two vertical
strands go either both over or both under the horizontal strand, the two Kauffman states
will have the same labelling for both codes. In the other case, the exponent of t will change
by ±1 for the code above, but by ±2 for the original code. It is not hard to see that the
signs are the same. Now apply the generalised clock theorem. 
Definition 4.5. We define the sign sgn(x) of a Kauffman state x of a connected
tangle diagram to be the sign of c(x) with h = −1, see definition 1.8.
Proof of proposition 4.2. Without loss of generality we may assume that at any
crossing of the diagram, all four regions are pairwise distinct. For, once we have shown the
above statement for this restricted case, it also holds true for any connected diagram, since
both sides are invariants of T up to isotopy.
We can regard the Kauffman states as a region-crossing assignment, so we can fix a map
P : K(T, s)→ Sn. If x and y are two Kauffman states in K(T, s), s ∈ S(T ), that are related
by a transposition move, x and y have opposite signs. On the other hand, P (x) and P (y)
also have opposite signs as elements of the permutation group. By the generalised clock
theorem, we know that any two Kauffman states in K(T, s) are connected by a sequence of
transposition moves. Hence sgn = ± sgn ◦P . The proposition now follows from the definition
of the determinant and lemma 4.4. 
Corollary 4.6. Let s be a site of a tangle T . Then
∇sT (1, . . . , 1) = 0⇔ H2(B3 r T,Bs) 6= 0.
Proof. Observe that the matrix A in (2) is obtained from the matrix A˜ in (3) by setting
all colours equal to 1. 
Remark 4.7. Given two sites s and s′, Bs can be obtained from Bs′ by applying some
element of the mapping class group of ∂B3 r ∂T . This means that if we know ∇sT for a
fixed site s and all tangles T , we also know ∇s′T for all sites s′, up to normalisation. But why
should we restrict ourselves to those subspaces B of the punctured sphere ∂B3 r ∂T that
come from sites? In principle, we can consider H1(X˜L, B˜) for any such B and define ∇BT
to be the generator of the smallest principal ideal containing the first elementary ideal of
H1(X˜L, B˜). This gives rise to infinitely many polynomial invariants ∇BT for any given tangle
T . However, if for example H1(X˜L, B˜) has a presentation matrix with more columns than
rows, i. e. more generators than relations, then ∇BT will be zero. This is for example always
the case if χ(B) > (1− n) = χ(Bs).
Figure 13. A transposition move of Kauffman states
I.4. GEOMETRIC INTERPRETATION OF ∇ST 31
The above considerations give rise to the following proposition.
Proposition 4.8. Let B be an essentially embedded subsurface of a 2n-punctured sphere,
i. e. the map pi1(B) → pi1(∂B3 r ∂T ) induced by the inclusion is injective. Suppose further
that χ(B) = 1−n. Then there exists an element g in the mapping class group of ∂B3r ∂T
such that g(B) is homotopic to Bs for some site s.
Proof. We fix some cellular structure on XT with a single 0-cell such that B deforma-
tion retracts onto a 1-dimensional sub-cell-complex. Its 1-cells cannot be interleaved, so by
sliding their endpoints along each other, we can arrange that they are attached to the 0-cell
like petals. Since B is essentially embedded, (∂B3 r ∂T )r B is a disjoint union of (n+ 1)
discs, each of which contains at least one puncture. Hence, we can apply an element of the
mapping class group of the punctured sphere to obtain a “standard” surface in a punctured
sphere which only depends on the partition of the punctures induced by B. Since any such
partition can be achieved by Bs for some site s, the result follows. 
Questions 4.9. In the context of the proposition above, there are some open questions
that we would like to answer; in particular:
• Are there any general relations between the polynomials ∇sT for different sites s, in
particular those that cannot be twisted into one another by applying some element
of the mapping class group?
• Can we compute the action of the mapping class group of ∂B3 r ∂T on ∇sT ?
• What is the best way to normalise ∇sT,new?
• Can we describe higher elementary ideals of H1(X˜T , B˜s)? What about ∇BT for the
case χ(B) < 1− n?
• Is there a geometric interpretation of the glueing formula from proposition 1.15, for
example via some Mayer-Vietoris argument?
Comparison with other definitions of Alexander polynomials for tangles. In
the introduction to this chapter, we mentioned several other generalisations of the Alexander
polynomial to tangles. In the remainder of this section, we try to compare our invariant to
some of the other definitions. I hope to come back to this at some point in the future to
make some of those vague statements below more precise.
Archibald’s invariant. In [Arc10], Archibald defines her polynomial Alexander invari-
ant(s) for tangles directly via Alexander matrices. However, she uses a different cellular
decomposition of the tangle complement to calculate the Alexander matrix; ours comes
from the Dehn representation of the fundamental group of the complement, hers from the
Wirtinger representation. For the latter, one fixes a single 0-cell above a tangle digram and
adds a loop for each connected arc in the diagram, as illustrated below.
b
32 I. ALEXANDER POLYNOMIALS FOR TANGLES
These loops generate the fundamental group of the tangle complement. One then adds 2-
cells, one for each crossing, which gives the relations between the generators. The columns of
the Alexander matrix correspond to crossings as in our construction, but the rows correspond
to arcs. Archibald then considers square matrices obtained by deleting some rows that
correspond to arcs that meet the boundary, which we call “marked” arcs. A simple counting
argument shows that we need to delete exactly n rows (if there are no closed arcs).
Let us assume for simplicity that every arc meets the boundary at most once. (We can always
find such a diagram for a given tangle.) Then, those square matrices define a presentation
of H1(X˜T , B˜), where B is the subspace of the boundary given by the 0-cell and those 1-cells
corresponding to marked arcs. This subspace B does not necessarily have to be homotopic
to one that comes from a site in our construction. However, after applying some element
of the mapping class group of the boundary, it is, like in the last step of the proof of
proposition 4.8. Thus, we can calculate Archibald’s invariant from our invariants and vice
versa, up to normalisation.
Diagrammatic invariants. Using the skein relation for the single-variate Alexander
polynomial and the convention that any diagram with an unknot component is zero, one
can reduce any given tangle to a linear combination of “simpler” tangles. For a suitable
(minimal) choice of such “elementary” tangles, this also gives an invariant. (This can be
easily seen by adapting the arguments from [LM87].) Since ∇sT also satisfies the skein
relation, we can compute it by substituting the elementary tangles by their polynomials ∇sT .
Computations suggest that one can also go the other direction. Since Bigelow’s [Big12] as
well as Polyak’s [Pol10] invariants also satisfy the skein relation, this would imply that all
of these polynomials contain basically the same information.
For the multivariate version, the same approach does not work. Computations suggest that
∇sT and Kennedy’s multivariate version [Ken12] of Bigelow’s invariant are closely related,
but I have been unable to make this relationship precise. If we restrict ourselves to 4-ended
tangles, we saw in proposition 3.1 that there is basically only one piece of information in ∇sT .
One can show that Kennedy’s invariant, consisting (a priori) of nine different polynomials,
contains at most two different pieces of information. It would be interesting to know if one
can make this result as strong as for ∇sT .
Sartori’s invariant. In [Srt13], Sartori interpreted the Alexander polynomial in terms
of the representation theory of gl(1|1). I assume that one can interpret the tangle Floer
homology defined in the next chapter as a special case of Petkova and Vértesi’s more gen-
eral combinatorial tangle Floer homology [PV14], namely their one-sided case. Since their
construction can be seen as a categorification of Sartori’s invariant [EPV15], this would then
imply a relationship between ∇sT and Sartori’s one-sided invariant, taking the form of a
map from a Uq(gl(1|1))-representation V to C(q). Under this map, ∇sT (q) should probably
be the image of some element in V that corresponds to the site s in Petkova and Vértesi’s
construction.
CHAPTER II
A Heegaard Floer homology for tangles
In this chapter, we categorify the polynomial invariants ∇sT from the first chapter. We
start in section II.1 by defining a tangle Floer homology ĤFT in terms of Juhász’s sutured
Floer homology SFH [Juh06a]. Using our geometric interpretation of the polynomial tangle
invariants ∇sT from section I.4 and a description of the decategorification of SFH due to
Friedl, Juhász and Rasmussen [FJR09], we show that ĤFT categorifies ∇sT . In section II.2,
we give an independent, but equivalent construction of ĤFT from Heegaard diagrams for
tangles. The main advantage over the first definition via sutured Floer homology is that
we naturally get relative gradings for all sites simultaneously. For the sutured approach,
this would require a means of comparing Spinc-structures for different sets of sutures, see
remark 2.24.
In order to obtain a glueing theorem for ĤFT that categorifies the glueing formula for ∇sT
(proposition I.1.15), we need to add some extra structure to ĤFT. This is done in section II.3,
using Zarev’s bordered sutured Floer theory. Note that in the first two sections, we are
working over the coefficient ring Z, whereas in the third, we restrict to Z/2-coefficients.
II.1. A categorification via sutured Heegaard Floer theory
In [Juh06a], Juhász generalised the hat version of Heegaard Floer homology of closed
three manifolds and links to balanced sutured manifolds, certain manifolds with boundary
together with some extra structure on the boundary. He used this sutured Floer homology,
denoted by SFH, to give short proofs of a number of known results, e. g. that link Floer
homology detects the Thurston norm and fibredness. Juhász also proved a surface decompo-
sition formula, which says that SFH behaves very nicely under splitting a balanced sutured
manifold along certain embedded surfaces. For all basic definitions and properties of SFH,
we refer the reader to Juhász’s original papers [Juh06a, Juh06b, Juh08] and Altman’s intro-
ductory article [Alt13].
In this section, we give a quick definition of a tangle Floer homology ĤFT in terms of SFH
and show that its Euler characteristic agrees with the polynomial invariant∇sT . Then, we use
a version of Juhász’s surface decomposition formula to prove symmetry relations for ĤFT.
Definition 1.1. With an r-component tangle T and a site s of T , we associate a
sutured 3-manifold XsT defined as follows: The underlying 3-manifold with boundary is
XT = B
3 r ν(T ), the complement of a tubular neighbourhood of T in B3. The sutures
on ∂XsT are obtained by placing two oppositely oriented meridional circles around closed
components of the tangle and meridional circles around the ends of the open components
and performing surgery along the arcs in s, see figure 14. We orient the sutures such that
one component of R− is contained in the boundary of the 3-ball.
33
34 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
a
b
c
d
T
Figure 14. The set of sutures (green curves) on XsT for a 4-ended tangle T and
s = d. Any closed components of T get two meridional sutures as in the case of
knots and links.
Theorem 1.2. The sutured Floer chain complex SFC(XsT ) is an invariant of the tangle T
and the site s up to chain homotopy equivalence. Furthermore
(4) χ(SFC(XsT ))(t
2
1, . . . , t
2
r) =˙
∏
(c− c−1) · ∇sT (t1, . . . , tr),
where =˙ denotes equality up to multiplication by a unit and the product on the right is over
all closed components of T and their colours c. For a 2-ended tangle T , SFC(X∅T ) agrees
with ĈFL(L), where L denotes the link obtained by joining the two open ends of T .
Definition 1.3. We denote the chain complex SFC(XsT ) by ĈFT(T, s) and its homology
by ĤFT(T, s), the tangle Floer homology of T at s.
Definition 1.4. The sutured Floer homology of any sutured manifold (M,γ) comes
with two gradings: a grading by relative Spinc-structures of (M,γ) and, for each such Spinc-
structure, a relative homological Z/2-grading. Spinc-structures of (M,γ) form an affine
space of H1(M). So if M = XsT for an r-component tangle T , an orientation on T induces
a relative Zr-grading, which we call the Alexander grading. The sutured Floer chain
complex of any (M,γ) splits along Spinc-structures. For our tangle Floer homology this
means that we can write
ĈFT(T, s) =
⊕
a∈Zr
ĈFT(T, s, a),
where ĈFT(T, s, a) denotes the component of ĈFT(T, s) in a fixed Alexander grading a.
Remark 1.5. In the next section, when we study Heegaard diagrams of tangles, we
see how to lift the homological Z/2-grading to a relative Z-grading and how to define the
Alexander grading for all sites simultaneously. To achieve this purely in terms of sutured
Floer homology, one would have to relate relative Spinc-gradings corresponding to different
sets of sutures, see remark 2.24.
Proof of theorem 1.2. We first check that XsT is balanced. Say T has n open com-
ponents and without loss of generality, we may assume that there are no closed components.
The site s consists of (n−1) open regions, so there are (n−1) arcs which we have performed
surgery along. Hence, R− is a sphere with (n + 1) punctures, so it has Euler characteristic
(1− n). Each annulus around an open tangle component contributes 0 to the Euler charac-
teristic, but each surgery decreases the Euler characteristic by 1.
II.1. A CATEGORIFICATION VIA SUTURED HEEGAARD FLOER THEORY 35
Obviously, XsT is an invariant of T , and so is its sutured Floer homology. Therefore, it
only remains to check the identity (4). For this, we use the explicit formula from [FJR09,
proposition 5.1] for the graded Euler characteristic χ(SFH(M,γ)) of a sutured manifold
(M,γ): Consider the pair (M,R−) and its maximal Abelian cover (M˜, R˜−). Let A be a
square presentation matrix of H1(M˜, R˜−) as a H1(M,R−)-module. Then χ(SFH(M,γ)) is
equal to det(A). We essentially did all the work in the previous section. First, suppose
T does not have a closed component. Then, using the notation from proposition I.4.2,
H1(X
s
T , R−) ∼= H1(B3 r T,Bs) and the same holds for the maximal Abelian covers, so we
are done by the same proposition. For the general case, we get a factor (c − c−1) for each
closed component by observation I.4.3, noting that we need to take the square of all variables
to match the convention used in the definition of ∇sT .
Finally, specialising to 2-ended tangles T representing a link L, we observe that X∅T is simply
the complement of a tubular neighbourhood of L in S3 with two meridional sutures on each
boundary component, so SFH(X∅T ) agrees with the ordinary link Floer homology ĤFL(L)
by [Juh06a, proposition 9.2]. 
The following results can be regarded as the categorification of their counterparts in
sections I.2 and I.3.
Proposition 1.6. Let m(T ) denote the mirror image of a tangle T . Let ĈFT∗(T, s)
denote the dual chain complex of ĈFT(T, s), with the usual convention that all gradings are
reversed. Then
ĈFT(m(T ), s) ∼˙= ĈFT∗(T, s),
where ∼˙= denotes graded chain homotopy equivalence up to an overall shift of the gradings.
Proof. This follows from the previous theorem and [FJR09, proposition 2.14]. 
Proposition 1.7. Let T be an oriented r-component tangle and s a site of T . If
r(T, t1) denotes the same tangle T with the orientation of the first strand reversed, then for
all Alexander gradings a = (a1, . . . , ar) ∈ Zr,
ĈFT(r(T, t1), s, a) ∼˙= ĈFT(T, s, (−a1, a2, . . . , ar)).
Similarly, if r(T ) denotes the tangle T with the orientation of all strands reversed, then
ĈFT(r(T ), s, a) ∼˙= ĈFT(T, s,−a).
Proof. The orientation of a tangle component is just a choice of an orientation of its
meridian. This does not affect the relative homological grading. 
Proposition 1.8. Let T be a 4-ended tangle. We distinguish between the same two cases
as in proposition I.3.1:
c
d
a
b
p
p
q
q
case I
c
d
a
b
p
p
q
q
case II
36 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
In both cases,
(B-D) ĈFT(T, b) ∼˙= ĈFT(r(T ), d).
In case I, we also have
(A-C) Vp ⊗ ĈFT(T, a) ∼˙= Vq ⊗ ĈFT(r(T ), c),
where Vt denotes a 2-dimensional vector space supported in consecutive Alexander and ho-
mological gradings. In case II (and in case I with p = q), the second identity holds if we
drop the tensor factors Vp and Vq.
Proof. Let us consider relation (B-D) first. The underlying sutured manifolds are the
same after switching the roles of R− and R+ on one side. This can be easily seen by pushing
the two meridional sutures of the open tangle components through the tangle. Then, by
[FJR09, proposition 2.14], the sutured Floer homologies are identical, except that the Spinc-
gradings are opposite to each other. Now apply proposition 1.7.
In case II, (A-C) (without the tensor factors) follows from the same arguments. In case I,
relation (A-C) is an exercise in applying the surface decomposition formula for sutured Floer
homology, see figure 15. Consider XcT . Let N be a tubular neighbourhood of the union of R−
and the component of R+ corresponding to the q-strand. Let S be the surface obtained as the
intersection of N with the closure of X ′T := X
c
T rN . Note that X ′T is diffeomorphic to XT .
We turn it into a balanced sutured manifold by adding a single suture on the boundary of
the 3-ball, separating the p-ends from the q-ends. We get a decomposition
XcT  S X ′T ∪N,
which satisfies the conditions of [Juh06b, proposition 8.6]. Thus,
SFH(XcT ) = SFH(X
′
T )⊗ SFH(N).
It is now straightforward to calculate SFH(N) which gives Vq. Thus,
ĤFT(T, c) = SFH(X ′T )⊗ Vq
and similarly
ĤFT(T, a) = SFH(X ′′T )⊗ Vp,
a
b
c
d
T
(a) XcT (b) N
a
b
c
d
T
(c) X ′T
Figure 15. The surface decomposition used in the proof of proposition 1.8
II.2. HEEGAARD DIAGRAMS FOR TANGLES 37
where X ′′T agrees with X
′
T , except that the roles of R− and R+ are interchanged. To get
relation (A-C), we now argue just as before. 
II.2. Heegaard diagrams for tangles
Throughout this section, let T be an oriented tangle with n open and m closed compo-
nents and s a site of T . The complement of a tubular neighbourhood of the tangle in the
closed 3-ball is denoted by XT . In the following, we adapt the Heegaard Floer construction
for knots and links [Ras03, OS03a, OS05] to tangles, which gives us another, but equivalent
definition of ĤFT to the one defined in the previous section.
Definition 2.1. A Heegaard diagram for a tangle consists of the following data:
• An oriented surface Σg of genus g with 2(n + m) boundary components, denoted
by Z, which are partitioned into (n+m) pairs,
• a set αc of (g +m) pairwise disjoint circles α1, . . . , αg+m on Σg,
• a set αa of 2n pairwise disjoint arcs αa1, . . . , αa2n on Σg which are disjoint from αc
and whose endpoints lie on Z, and
• a set β of (g +m+ n− 1) pairwise disjoint circles β1, . . . , βg+m+n−1 on Σg.
We write α := αc ∪αa and impose the following conditions on the data above:
• Contracting all boundary components turns αa into a single circle. So in particular,
on each component of Z, there are either no or exactly two endpoints of αa.
• The surface Sαc(Σg) obtained by surgery along the curves in αc is a disjoint union of
m annuli, each of whose boundary is a pair in Z, and a 2-sphere with 2n boundary
components. We denote the set of circles in Z which meet the α-curves by Zα.
• The surface Sβ(Σg) obtained by surgery along the curves in β is a disjoint union
of (n+m) annuli, each of whose boundary is a pair in Z.
Remark 2.2. We can recover the tangle complement XT from this data by attaching
2-handles to Σg × [0, 1] along αc × {0} and β × {1}; the α-arcs then correspond to S1 from
definition I.1.1. Conversely, we can pick a self-indexing Morse function f on B3 which is
identical to 32 on a neighbourhood of S
1 and which has a single minimum and a single maxi-
mum on each closed component of T and a single minimum on each open tangle component.
Then, the Heegaard surface is equal to f−1(32) modulo punctures at the tangle ends; the α-
and β-circles are the loci of points on this surface flowing from/to the index 1 and 2 critical
points, respectively. Our convention on the orientation of the Heegaard surface is
that its normal vector field (using the right-hand rule) points in the positive direction of the
Morse function, i. e. in the direction of the β-curves. However, we usually draw the Heegaard
surfaces such that this normal vector field points into the plane.
Remark 2.3. Given a tangle T , we can endow XT with the structure of a bordered
sutured manifold as follows: Each closed component gets two oppositely oriented meridional
circles and around each tangle end, we have a single suture such that the boundary of B3
minus a neighbourhood of the tangle ends lies in R−. Furthermore, the arcs S1r∂T together
with small neighbourhoods of the endpoints on Z constitute the arc diagram. Similarly,
Heegaard diagrams for tangles can be viewed as bordered sutured Heegaard diagrams, see
section II.3, in particular definition 3.5 and figure 21a.
38 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
Lemma 2.4. Every tangle T has a Heegaard diagram. Moreover, any two diagrams for
the same tangle can be obtained from one another by a sequence of the following moves:
• an isotopy of an α- or β-circle or an isotopy of an α-arc relative to its endpoints,
• a handleslide of a β-circle over another β-circle,
• a handleslide of an α-curve over an α-circle and
• stabilisation.
Proof. In view of remark 2.3, this is a special case of [Zar09, proposition 4.5]. The first
part also follows from the next example. 
Example 2.5. For a 1-crossing tangle, we draw the Heegaard diagram shown in fig-
ure 16b. From this, we can obtain a Heegaard diagram for any tangle without closed com-
ponents as follows: We cut a tangle diagram of a given tangle up into 4-ended tangles with a
single crossing each. Then, for each such component, we can use our Heegaard diagram from
figure 16b and then glue these copies together along Z according to the tangle diagram. For
tangles with closed components, we can do the same except that into each closed component,
we insert a copy of the “ladybug” from figure 16c.
Example 2.6. For rational tangles, we can draw Heegaard diagrams on genus 0 surfaces.
As illustrated in figure 17, this can be seen by performing Dehn twists on the Heegaard
diagram for the 1-crossing tangle in figure 16b. In fact, a 4-ended tangle is rational iff it has
a genus 0 Heegaard diagram. Indeed, a genus 0 Heegaard diagram for a 4-ended tangle has
no α-circles and just a single β-circle. By definition, we know that performing surgery along
this β-circle gives us two cylinders, so it separates two punctures from the other two.
In the following, let H = HT = (Σg,Z,α,β) be a Heegaard diagram for T .
Definition 2.7. Let T = TH be the set of tuples x = (x1, . . . , xg+m+n−1) of points
x1, . . . , xg+m+n−1 ∈ α ∩ β such that there is exactly one point xi on each α- and β-circle,
and at most one point on each α-arc. T will be the generating set of the chain module
defined later on, so we call the elements in T generators.
Following the notation from definition I.1.8, a site s ∈ S(T ) corresponds to an (n − 1)-
element subset of αa. With each generator x ∈ T, we associate the site s(x) consisting of
(a) A single crossing tangle
b
b
b
b
αc
β
Z
D
C
B
A
(b) A Heegaard diagram for
the tangle on the left
bb
bb
(c) A ladybug, see also
[BL11, figure 8]
Figure 16. Two building blocks of tangle Heegaard diagrams for example 2.5
II.2. HEEGAARD DIAGRAMS FOR TANGLES 39
(a) A rational tangle obtained from the 1-crossing tangle
in figure 16a by three Dehn twists on the right
b
b
b
b
b
b
b
b
b
b
(b) A genus 0 Heegaard dia-
gram for the tangle in (a)
Figure 17. Rational tangles have genus 0 Heegaard diagrams.
all those α-arcs that are occupied by an intersection point in x. We denote the set of all
generators corresponding to a given site s by Ts. Thus, we obtain a partition
T =
∐
s∈S
Ts.
We define D to be the free Abelian group generated by the connected components of Σg r
(α ∪ β ∪ Z), which we call regions. In other words,
D = DH := H2(Σg,α ∪ β ∪ Z).
Elements of this group are called domains. Given two points x,y ∈ T, we define pi2(x,y)
to be the subset of those domains φ which satisfy
d(dφ ∩ β) = x− y.
We call elements in pi2(x,x) periodic domains. Note that this does not depend on the
choice of x ∈ T. Furthermore, let
pi∂2 (x,y) := {φ ∈ pi2(x,y)| Z ∩φ = ∅}.
A Heegaard diagram is called admissible if every non-zero periodic domain in pi∂2 (x,y) has
positive and negative multiplicities.
Lemma 2.8. Every tangle diagram can be made admissible by isotopies of β. Further-
more, any two such diagrams for the same tangle can be transformed into one another by a
sequence of Heegaard moves from lemma 2.4 through admissible diagrams.
Proof. In light of remark 2.3, this is a special case of [Zar09, proposition 4.11 and
corollary 4.12]. Note that our terminology differs slightly from Zarev’s: He does not include
regions near basepoints in pi2(x,y), so in our special case, his definition of pi2(x,y) coincides
with our pi∂2 (x,y). Thus, the distinction in his terminology between periodic and provincial
periodic domains becomes irrelevant. 
Lemma 2.9. There is an isomorphism pi2(x,x) ∼= H2(XT ,Z ∪αa;Z) ∼= Zn+2m+1. Fur-
thermore, pi∂2 (x,x) = Zm; in particular, any Heegaard diagram for a tangle without closed
components is admissible.
40 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
Proof. By attaching 2-handles to the α- and β-circles we see that
pi2(x,x) ∼= H2(XT ,Z ∪αa;Z).
This group is generated by annuli, one for each open and two for each closed component,
and two discs, one at the front and the other at the back, the only relation being that the
sum of the annuli is equal to the sum of the discs. This proves the first statement.
The second follows from the first: Let us write a given periodic domain in pi∂2 (x,x) as a
linear combination of the basic periodic domains described above such that the coefficient
of one of the two discs, say, is zero. Then the coefficients of the two annuli of each closed
component must be opposite and all other coefficients must be zero. In fact, the differences
of the two annuli for each closed component form a basis of pi∂2 (x,x). 
Lemma 2.10. pi2(x,y) is non-empty for all pairs (x,y) ∈ T2.
Proof. For any pair (x,y) ∈ T2, there exists a 1-cycle γ in C1(α ∪ β ∪ Z) such that
d(γ ∩ β) = x − y. Indeed: First, we choose a 1-chain on C1(β) with this property. Then,
since there is exactly one intersection point on every α-circle for each x and y, we can add
1-chains in C1(αc) such that the boundary of the new 1-chain lies on the α-arcs only. But
since γ is allowed to have Z-components, we can get rid of these intersection points, too,
and obtain our cycle γ.
Next, we can add Z-cycles to γ such that the resulting 1-cycle is 0 in H1(XT ). Adding α- and
β-cycles gives us another 1-cycle γ′ which is 0 in H1(Σg) and also satisfies d(γ′∩β) = y−x.
So we are done. 
Our next goal is to define a relative Alexander grading on generators. We do this by
counting Z-components of domains which connect two generators.
Definition 2.11. By definition, the endpoints of the α-arcs divide each circle in Zα into
two components, which we suggestively call front and back component. Let Σfg , (α∪β∪Z)f
and αf denote the spaces obtained from Σg, (α∪β ∪Z) and α respectively by contracting
the back component of each circle in Zα to a point. Note that β ∩ Z = ∅, αc ∩ Z = ∅ and
that the images of α-arcs become a single circle in αf . Let f : (α∪β∪Z)→ (α∪β∪Z)f be
the quotient map and ∂ : D → H1(α∪β ∪Z) the boundary map of the long exact sequence
of the pair (Σg,α ∪ β ∪ Z). Now, consider the following diagram, where ι, ιf , i and j are
induced by inclusions, and pr2 is the projection onto the second summand:
D H1(α ∪ β ∪ Z) H1(Σg) H1(Σg)/〈αc,β〉 ∼= H1(XT )
H1((α ∪ β ∪ Z)f ) H1(Σfg )
H1(α
f ∪ β)⊕H1(Zf ) H1(Zf )
∂

ι
f
∼=
j
ιf
∼=
jf

pr2
c=jf◦ii
II.2. HEEGAARD DIAGRAMS FOR TANGLES 41
Note that the orientation of T induces an orientation of the meridians using the right-hand
rule, which gives rise to a canonical identification H1(XT ) ∼= Zn+m. We define
Af : D → H1(XT ) ∼= Zn+m
as the composition c ◦ pr2 ◦ f ◦ ∂. Similarly, by contracting the front components of Zα, we
obtain a homomorphism
Ab : D → Zn+m.
Lemma 2.12. Af and Ab are constant on pi2(x,y) for all pairs (x,y) ∈ T2.
Proof. It suffices to show that Af and Ab vanish on periodic domains. But this is
obvious from the description of the periodic domains in (the proof of) lemma 2.9. The
annuli have cancelling Z-components of the corresponding tangle component and the Z-
components of the two discs are the sums of all front/back Z-components. 
Combining lemmas 2.10 and 2.12 enables us to define a relative grading on T.
Definition 2.13. The homomorphism A = Af + Ab : D → Zn+m induces a relative
grading A : T→ Zn+m by setting
A(y)−A(x) = A(φ) for φ ∈ pi1(x,y).
We call A theAlexander grading. Let Ar be the composition of A with the map Zn+m → Z
that adds all components. (“r” stands for “reduced”.) We often introduce shifts by half-
integers to achieve a certain symmetry which mimics the normalisation of ∇sT ; thus the
Alexander grading appears to be a relative (12Z)
n+m-grading.
Lemma 2.14. pi∂2 (x,y) is non-empty for all pairs (x,y) ∈ T2, iff x and y are in the
same Alexander grading and belong to the same site.
Proof. The only-if part is clear. The opposite direction follows from a refinement of
the proof of 2.10: We can now get a 1-cycle γ in C1(α ∪ β) such that d(γ ∩ β) = x − y,
because the generators belong to the same site. This 1-cycle is already zero in H1(XT ),
since the generators are in the same Alexander grading. Then we might have to add α- and
β-cycles as before and we are done. 
To define the differentials in our chain complexes, we count the number of points in 1-
dimensional moduli spaces of holomorphic curves modulo an R-action. The formal dimension
of these moduli spaces is called the Maslov index. It can be computed combinatorially, as
shown in [Lip05, corollary 4.10]. We take this combinatorial formula as a definition.
Definition 2.15. Let φ ∈ pi2(x,y) for some x,y ∈ T. We define the Maslov index by
µ(φ) = e(φ) +mx(φ) +my(φ),
where e(φ) is the Euler measure of φ and mx(φ) and my(φ) are the multiplicities of φ at x
and y, respectively. More explicitly, given a region ψ of the Heegaard diagram, let mψ(φ)
denote the coefficient of ψ in φ. Then
e(φ) =
∑
regions ψ
mψ(φ)(χ(ψ)− 14#{acute corners of ψ}+ 14#{obtuse corners of ψ}).
42 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
Furthermore, for any x ∈ T, let
mx(φ) =
∑
xi∈x
mxi(φ),
where mx(φ) is the average of the mψi(φ) in the four quadrants ψ1, . . . , ψ4 at x.
Lemma 2.16. Given φ ∈ pi2(x,y) and ψ ∈ pi2(y, z), µ(φ) + µ(ψ) = µ(φ+ ψ).
Proof. This follows from basically the same arguments as [Srk06, theorems 3.1 and 3.3].
We give some details nonetheless. First of all, note that the Euler measure is additive. Hence,
all we need to show is that
mx(φ) +my(φ) +my(ψ) +mz(ψ) = mx(φ+ ψ) +mz(φ+ ψ).
This simplifies to
my(φ) +my(ψ) = mx(ψ) +mz(φ).
Theorem 3.1 from [Srk06] for n = i = 2, η1 = α and η2 = β gives us
mz(φ)−my(φ) = ∂φ · ∂β(ψ) and similarly
my(ψ)−mx(ψ) = ∂ψ · ∂β(φ),
where the product · denotes the “average” intersection number from [Srk06]. So we need to
see that
∂ψ · ∂β(φ) + ∂β(ψ) · ∂φ = 0.
The boundaries of the domains lie in α ∪ β ∪ Z. However, β ∩ Z = ∅, so the left-hand side
equals ∂α(ψ) ·∂β(φ)+∂β(ψ) ·∂α(φ) = ∂α∪β(ψ) ·∂α∪β(φ). To see that this is zero, we modify
the Heegaard surface by contracting all boundary components. Then the left-hand side is
equal to ∂(ψ) · ∂(φ), and this is indeed zero. 
Lemma 2.17. µ is constant on pi2(x,y) for all pairs (x,y) ∈ T2.
Proof. Applying the previous lemma to z = y, we see that all we need to show is that
µ(φ) = e(φ) + 2my(φ)
vanishes for all periodic domains φ ∈ pi2(y,y). In fact, it suffices to show this for every
elementary periodic domain φ from the proof of lemma 2.9. Consider φ as a subsurface
of Σg × {0} or Σg × {1}, depending on whether it corresponds to an annulus or disc from
performing surgery along α-circles or β-circles. The annulus or the disc is obtained from φ by
performing surgery along the respective circles or attaching single discs to them. The former
reduces the Euler measure of φ by 2, the latter by 1. For an annulus, this contribution
is cancelled by the term 2my(φ), since every α- and β-circle is occupied by exactly one
intersection point of y. So in this case, we have µ(φ) = µ(annulus) = 0. For a disc, we also
need to take into account that y occupies (n − 1) α-arcs. In this case, 2my(φ) contributes
(n− 1). On the other hand, the disc has 4n corners, so e(disc) = 1− n. 
Combining lemmas 2.16 and 2.17, we can now define a relative grading on generators
induced by the Maslov index µ, just as for the Alexander grading.
II.2. HEEGAARD DIAGRAMS FOR TANGLES 43
Definition 2.18. The δ-grading on generators is a relative 12Z-grading defined by
δ(y)− δ(x) = µ(φ), where φ ∈ pi2(x,y).
We also define a relative Z-grading, the homological grading, by
h(y)− h(x) = h(φ) := 12Ar(φ)− µ(φ), where φ ∈ pi2(x,y).
In short,
h = 12A
r − δ.
Lemma 2.19. The homological grading is well-defined.
Proof. The homological grading is a priori a relative 12Z-grading. However, it is clear
from the alternative formula for µ from [Srk06, section 2], that µ is an integer for a closed
Heegaard surface. In our case, we can easily obtain a closed surface by contracting the
boundary components as in the proof of lemma 2.16. Thus 2µ(φ) ≡ A(φ) mod 2 for any
φ ∈ pi2(x,y). 
Remark 2.20. When comparing these gradings with those in link Floer homology, note
that we are using a Heegaard surface with punctures instead of marked points. For two-
ended tangles, our conventions agree with those in [BL11, section 3.1, equations (3.2)–(3.4)],
noting that tangle ends that point into the 3-ball are represented by Xs and outgoing ones
by Os. Although we have three different gradings, we call their union the bigrading, since
any one of them is determined by the other two.
Definition 2.21. Let T be a tangle andHT a Heegaard diagram for T . We define a chain
module ĈFT(HT ) as follows. The underlying Z-module is freely generated by the elements
in T. It carries three gradings, the Alexander grading and the homological grading and the
δ-grading, induced by the gradings on generators. We write ĈFT(HT , s) for the submodule
generated by those elements in Ts and ĈFT(HT , s, a) for the submodule generated by those
elements in Ts of Alexander grading a ∈ Zn+m. The differential on ĈFT(HT ) is given by
∂x =
∑
y∈T
∑
φ∈pi∂2 (x,y)
µ(φ)=1
#M̂(φ) · y,
where #M̂(φ) denotes the signed count of holomorphic curves associated with φ, see for
example [Lip05]. Note that the sum is over domains in pi∂2 (x,y), i. e. those that avoid Z.
Since there are no such domains between generators in distinct Alexander gradings or sites,
the chain module ĈFT(HT ) admits a splitting into summands ĈFT(HT , s, a).
Theorem 2.22. The differential on ĈFT(T, s) is well-defined. Furthermore, the bigraded
chain homotopy type of (ĈFT(HT , s), ∂) is an invariant of the tangle T and it agrees with
ĈFT(T, s) from definition 1.3.
Furthermore, we can now easily re-prove the following result.
Lemma 2.23. The graded Euler characteristic of the chain module ĈFT(HT , s) coin-
cides with the polynomial invariant ∇sT up to normalisation and additional factors for closed
components as in theorem 1.2.
44 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
b
b
b
b
φ
ψ
φ′
ψ′
D
C
B
A
o
1
2u
1
2
o
1
2u−
1
2
h−1o−
1
2u−
1
2
o−
1
2u
1
2
δ
1
2
δ
1
2
(a) A positive crossing
o−
1
2u−
1
2
o
1
2u−
1
2
ho
1
2u
1
2
o−
1
2u
1
2
δ−
1
2
δ−
1
2
b
b
b
b
φ
ψ
φ′
ψ′
D
C
B
A
(b) A negative crossing
Figure 18. The gradings of the generators of the two 1-crossing tangles. The over-
strand is coloured by o and the under-strand by u. Compare this to the Alexander
codes from figure 6. Our conventions agree with [OS07, figure 5 and 6] and [BL11,
figure 10].
Remark 2.24. All three gradings on ĤFT are relative. One can probably fix an absolute
Alexander grading which agrees with the natural normalisation of the polynomial tangle
invariants. For example, as long as ∇sT 6≡ 0, we could simply ask for the Euler characteristic
of ĈFT(HT , s) to agree with ∇sT . For tangles in more general 3-manifolds, one could also
use an absolute Spinc-grading, or more explicitly, non-vanishing vector fields, see [HR11]
and [HR12]. For 4-ended tangles, we can cheat like we usually do when computing gradings
on ĤFK: We simply use the symmetry relations for opposite sites (proposition 1.8) to fix an
absolute Alexander grading such that the graded Euler characteristic agrees with ∇sT up to
multiplication by (±1).
Proof of lemma 2.23. We first calculate the gradings of the generators for the 1-
crossing diagrams, see figure 18. In each case, we have four connecting domains ψ, φ, ψ′
and φ′. The δ-grading of all these domains is +12 . This gives us the correct δ-grading on
generators, noting that the normal vector field of the Heegaard diagram, determined by
the right-hand rule, points into the plane. (For example, for the positive crossing, ψ is
in pi2(A,D) and the δ-grading increases along this domain by +12 .) Using the right-hand
rule convention from definition 2.11, we similarly obtain the correct Alexander gradings.
This determines the homological grading.
For a general tangle, we consider the Heegaard diagram induced by a tangle diagram as
discussed in example 2.5. Then, additivity of the gradings shows that the Alexander grading
of a generator in the whole diagram is the sum of the gradings in the local diagrams at
the crossings. For each closed component, we need to insert a ladybug into the Heegaard
diagram, see figure 16c; this multiplies the number of generators by two, since there are two
intersection points of the α-circle in a ladybug. It is straightforward to compute the grading
difference between corresponding generators of the two intersection points: the δ-gradings
agree and the Alexander gradings differ by 2. Hence, we get an extra factor (t− t−1) in the
decategorified invariant. 
Remark 2.25. For tangles without closed components, the Mathematica program
[APT.m] explicitly computes the generators of the categorified tangle invariant from a stan-
dard Heegaard diagram as in the proof above. For tangles with closed components, we need
to add the factor (t− t−1) for each closed component.
II.2. HEEGAARD DIAGRAMS FOR TANGLES 45
b
b
bb
b
b
b b
b
b
b
b
Figure 19. From a Heegaard diagram for a tangle to one for a sutured manifold,
illustrating the proof of theorem 2.22. This example shows the Heegaard diagram
for the 1-crossing tangle.
Proof of theorem 2.22. The theorem follows from the usual arguments in Heegaard
Floer homology. In fact, the identification of every component ĈFT(HT , s) of ĈFT(HT ) with
ĈFT(T, s) implies most of the result. The main idea is to modify our Heegaard diagram HT
in the following way, as illustrated in figure 19.
Let us consider a 2-torus with a fixed longitude and 2n disjoint meridians. Puncture the
torus 2n times along the longitude such that any two meridians are no longer homotopic. We
consider the remaining segments of the longitude as α-arcs and the meridians as β-circles.
Note that each β-circle intersects exactly one α-arc in a single point and there are exactly
2n connected components in their complement on the punctured torus. We place a puncture
in each of these components. Finally, we attach the (now 4n-punctured) torus to Σ in such
a way that each α-arc in the torus closes an α-arc in the Heegaard surface Σg for our tangle.
This gives us a sutured Heegaard diagram H consisting of a 2n-punctured surface Σ with
(g + 2n) α-circles and (g + 2n+ n− 1) β-circles. However, H is not balanced unless n = 1.
So, let us fix a site s. By definition, s is a set of α-arcs inH that are occupied by generators in
Ts. An α-arc in H corresponds to an α-arc in the punctured torus which in turn corresponds
to the β-circle that it intersects. Thus, a site s gives rise to a collection of β-circles on the
punctured torus. For each such circle βi, we pick a path γi between the two adjacent
punctures (the dashed line in figure 19) which intersects no α-circle and no β-circle except
βi. We delete βi and cut the surface along γi. This gives us a new sutured Heegaard diagram
which is balanced. Let us denote it by Hs.
Now observe that generators in Ts correspond to generators in Hs and that domains in H
that avoid Z correspond to domains in Hs that avoid the boundary. Furthermore, if we
started with an admissible Heegaard diagram H, then Hs is also admissible. Hence, the
sutured Floer homology SFH(Hs) is well-defined and identical to ĤFT(T, s). Furthermore,
if we fix a site s, the homological grading on ĈFT(HT , s) agrees (by definition) with the
one on ĤFT(T ). Similarly, the Alexander grading agrees with the relative H1(XT )-grading
46 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
q
q
p
p
a
b
c
d
(a) The (2,−3)-pretzel tangle
b
b
b
b
b
b
d
x1
x2
b
a2
a1
b
b
b
b
b
b
b
b
d′
y1
y2 y3
b′
c3
c2
c1
(b) A Heegaard diagram for the tangle on the left
site a site b site c site d
a1y1 : p
0q+3δ−
1
2 by1 : p
−1q+1δ0 x1c1 : p+1q+2δ−
1
2 ((((
(((dy1 : p−1q−3δ0
a1y2 : p
0q+1δ−
1
2 by2 : p
−1q−1δ0 x1c2 : p+1q0δ−
1
2 dy2 : p
+1q+1δ0
a1y3 : p
0q−1δ−
1
2 ((((
(((by3 : p−1q−3δ0 x1c3 : p+1q−2δ−
1
2 dy3 : p
+1q−1δ0
a2y1 : p
0q+1δ−
1
2 x1b
′ : p+1q−3δ−1 x2c1 : p−1q+2δ−
1
2 ((((
((((x1d′ : p+1q+3δ−1
a2y2 : p
0q−1δ−
1
2 ((((
((((x2b′ : p−1q−3δ−1 x2c2 : p−1q0δ−
1
2 x2d
′ : p−1q+3δ−1
a2y3 : p
0q−3δ−
1
2 x2c3 : p
−1q−2δ−
1
2
(c) A table of the generators of ĤFT for the pretzel tangle above and their gradings. The
generators that can be cancelled are crossed out.
Figure 20. The calculation of ĤFT for the (2,−3)-pretzel, see example 2.26
on ĤFT(T ) induces by the Spinc-grading (see for example [Juh06a, definition 4.6]). It now
only remains to check that ĈFT(T ) is an invariant as a relatively graded complex for all sites
simultaneously. However, this follows from the observation that the Heegaard moves from
lemma 2.4 do not change the Alexander nor the δ-grading of the generators that correspond
to one another under these moves. 
Example 2.26 (the (2,−3)-pretzel tangle). In figure 20, we compute our tangle Floer
homology for the (2,−3)-pretzel tangle. The shaded regions in the Heegaard diagram in
figure 20b show the only two domains that contribute to the differential. It is interesting
to note that if we set t := p = q, then the result for the sites a and c are the same and for
the sites b and d are the same after reversing the orientation t↔ t−1. For invariance under
mutation by rotating the tangle by pi in the plane, we need however b = d, which is only
true for the δ-graded invariant; see theorem 3.14 and example III.1.18.
II.3. Glueing via bordered sutured Floer theory
The tangle Floer homology defined in the previous section does not satisfy any glueing
formula because we only record domains away from the boundary. In this subsection, we use
bordered sutured Floer theory, developed by Zarev in [Zar09], to add such a glueing structure
to the invariant. We will assume some familiarity with [Zar09] and only give a short review
of the basic geometric objects. In particular, we will not give a complete definition of Zarev’s
invariants.
II.3. GLUEING VIA BORDERED SUTURED FLOER THEORY 47
Definition 3.1. A sutured surface is a quadruple (F,Λ, S+, S−), where F is a surface
with boundary and no closed components and Λ is a set of finitely many points on ∂F that
partition ∂F into two subsets S+ and S−. If
pi0(Λ)→ pi0(∂F )
is surjective, we call a surface non-degenerate, otherwise degenerate.
Definition 3.2. An arc diagram Z is a triple (Z,a,M), where Z is a set of oriented
line segments, a an even number of points on Z and M a matching of points in a. The
graph G(Z) of an arc diagram Z is the graph obtained from the line segments Z by
adding an edge between matched points in a. A parametrisation of a sutured surface
F is an embedding of a graph of an arc diagram Z into F such that the line segments Z are
mapped onto S+ and the image G(Z) is a deformation retraction of F .
Definition 3.3. A bordered sutured manifold is a quintuple (Y,Γ, F,Z, φ), where
Y is a sutured manifold with sutures Γ, F is contained in the closure of R− and
(F, ∂(Γ ∩ ∂F ),Γ ∩ ∂F,R− ∩ ∂F )
is a sutured surface parametrised by the arc diagram Z via an embedding φ : G(Z) ↪→ F
such that
(5) pi0(Γr F )→ pi0(∂Y r F )
is surjective.
Remark 3.4. Condition (5) is called homological linear independence. If we drop
this condition, Zarev’s invariants fail to be well-defined in general. Note that unlike Zarev, we
allow the sutured surfaces of bordered sutured manifolds to be degenerate. This allows us to
consider more general bordered sutured manifolds. If we restrict to non-degenerate sutured
surfaces, homological linear independence is automatically satisfied, see [Zar09, proposi-
tion 3.6].
Bordered sutured invariants and glueing. Given a bordered sutured manifold Y ,
Zarev defines two different invariants: a so-called type A structure B̂SA(Y ) and a type D
structure B̂SD(Y ). The former is an A∞-module over an algebra A(Z) associated with the
arc diagram parametrising the sutured surface on ∂Y ; the latter is a type D module over
this algebra. For a discussion of type A and D structures as algebraic objects, we refer the
reader to appendix A, in particular examples A.9 and A.13.
Given a (balanced) sutured manifold (Y,Γ) and a surface F in Y that has no closed compo-
nents and splits Y into two components Y1 and Y2 such that every component of F intersects
Γ non-trivially, we can endow Y1 and Y2 with the structure of bordered sutured manifolds
by choosing a parametrisation of F by an arc diagram. Then by [Zar09, theorem 1], the
sutured Floer complex SFC can computed as a certain tensor product  between the type A
and type D structures:
(6) SFC(Y ) = B̂SA(Y1) B̂SD(Y2).
48 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
Again, for details on the box tensor product , see appendix A, in particular definition A.17.
Unsurprisingly, the two structures B̂SA(Y1) and B̂SD(Y2) are defined in terms of Heegaard
diagrams for bordered sutured manifolds.
Definition 3.5. AHeegaard diagram of a bordered sutured manifold is obtained
from a Heegaard diagram of the underlying sutured manifold by adding the graph of the arc
diagram to it. To be more precise, consider a Heegaard diagram of the underlying sutured
manifold. Then we can embed the graph G(Z) into R− in such a way that it misses the
2-handles corresponding to the α-curves, simply by sliding them off those 2-handles. This
gives us an embedding of G(Z) into the Heegaard surface such that its image does not
intersect the α-curves. We view the images of the edges connecting points in a as α-arcs.
The image of Z lies on the boundary of the Heegaard diagram, i. e. the sutures, which we
usually draw in green. We put a marked point, a basepoint, in every open component of
the boundary minus the image of Z.
Bordered sutured manifolds for tangles. We can view Heegaard diagrams for tan-
gles (definition 2.1) as bordered sutured Heegaard diagrams, as explained in remark 2.3. In
this case, each of the line segments Z of the arc diagrams have only one marked point in a, so
the glueing surface is just a union of discs. However, when we pair two tangle complements
together to obtain a link complement, we need to glue along the whole boundary of the 3-
balls (minus the tangle ends), so we need to change the arc diagrams slightly. Depending on
whether we want to compute a type A or type D invariant, we use slightly different bordered
sutured structures on the tangle complements.
Definition 3.6. Consider the local picture 21a around a tangle Heegaard diagram from
definition 2.1. Note that we have added the two basepoints (the two dashes) on the original
suture. A type α end is obtained by removing one basepoint and adding an α-arc on the
opposite side as shown in figure 21b, which we call a silly arc. A type αβ end is obtained
from a type α end by capping off the suture by a β-circle and puncturing the region enclosed
by the silly α-arc and the new β-circle, see figure 21c.
Glueing type α and type αβ ends. Figure 21d illustrates how a type α and a
type αβ end locally glue together along the arc diagram to form a Heegaard diagram for the
glued manifold with two meridional sutures around the glued strand. Note that replacing
the original tangle end by a type α end does not change the generators of the Heegaard
diagram. For a type αβ end, the generators do change. However, when we glue a type α end
to a type αβ end, the new α-circle only intersects the β-circle from the type αβ end. So any
generator of the type αβ side that does not occupy the silly α-arc is killed in the pairing. If
we calculate the type A structure, we can indeed forget those generators and all arrows to
and from it, and the bordered sutured pairing formula from equation (6) still remains true.
For the type D side, this does not work in general, because in the box tensor product , we
also need to consider chains of arrows on the type D side that could possibly go via some of
the generators we want to forget (see definition A.17). We will avoid these complications by
only taking α ends for the type D structure and αβ ends for the type A structure.
II.3. GLUEING VIA BORDERED SUTURED FLOER THEORY 49
(a) An original tangle end (b) A type α end (c) A type αβ end
(d) A diagram before cutting/after glueing. It is a ladybug from figure 16c.
Figure 21. The bordered sutured Heegaard diagram around the tangle ends. To
get from (d) to (b) and (c), we cut along the dashed green line.
Unfortunately, when we glue two tangles together to obtain a knot or link, we cannot treat
all tangle ends in the same way. This is for two reasons: First, the arcs need to parametrise
the glueing surface, which in this case should be a 2n-punctured 2-sphere. Second, the
bordered sutured manifold should satisfy the homological linear independence condition (5).
Therefore, we have to remove some arcs and for this, we have lots of equally good or bad
choices.
Glueing tangle complements. We introduce some notation for the next theorem:
Let s be a site of a tangle T and XT the corresponding sutured manifold from section II.1.
Let XT1 and XT2 be the complements of two tangles T1 and T2 obtained by splitting XT
along some plane meeting l tangle components, see figure 22. We can turn XT1 and XT2
into two bordered sutured manifolds as follows: First of all, the glueing surface is obviously
the intersection of the cutting surface with XT . It is parametrised by those arcs connecting
the new tangle ends that we implicitly chose in defining T1 and T2. We replace the original
tangle ends by type α ends on XT2 and type αβ ends on XT1 . Next, we choose the sutures
on XT1 and XT2 to agree with those on XT on their respective common boundary. Finally,
if a suture cannot be homotoped away from the boundary of the cutting surface, we close
and connect it with the remaining arcs from the parametrisation of the glueing surface, as
illustrated in the upper part of figure 22.
By construction, if we glue XT1 and XT2 together, we reobtain the original sutured manifold,
except that there is now an extra pair of meridional sutures at those points of the tangles
50 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
T1 T2
Figure 22. An illustration for the glueing theorem 3.7
where we glue them together. We denote this new sutured manifold by X ′T . Let A be the
algebra corresponding to the glueing surface and I the corresponding ring of idempotents.
Let I ′ ⊂ I be the subring generated by those idempotents I(s) where s ranges over those
sets of arcs that contain the silly α-arcs at the α-ends. In other words, I ′ consists of those
idempotents that could belong to generators of B̂SD(XT2). We are now ready to state
the glueing theorem which can be viewed as a categorification of the glueing formula from
proposition I.1.15.
Theorem 3.7. With the notation from above, there exist vector space isomorphisms⊕
s2∩O2(T )=s∩O2(T )
ĈFT(T2, s2) −→ B̂SD(XT2)⊕
s1∩O1(T )=s∩O1(T )
ĈFT(T1, s1)⊗ V ⊗l −→ B̂SA(XT1).I ′⊕
s1∩s2=∅
(s1∪s2)∩O(T )=s
ĈFT(T1, s1)⊗ ĈFT(T2, s2)⊗ V ⊗l −→ ĈFT(T, s)⊗ V ⊗(l−c)
where O(T ) denotes the set of open regions of the tangle T , O1(T ) = O(T ) r O(T2) and
O2(T ) = O(T ) r O(T1), V is a vector space with two generators in grading t and h−1t−1
and c is the difference between the number of closed tangle components before and after the
splitting. The first map is a chain isomorphism after setting all algebra elements with moving
strands equal to zero. Furthermore, if one of B̂SA(XT1).I ′ and B̂SD(XT2) is bounded,
(7) ĈFT(T, s)⊗ V ⊗(l−c) = B̂SA(XT1).I ′  B̂SD(XT2).
Remark 3.8. The bordered sutured invariants for the (2,−3)-pretzel tangle that we
compute in theorem 3.14 are both bounded, so we can apply the glueing statement above.
Moreover, the invariants that we compute in proposition 3.10 for the (positive) rational
tangles are all bounded, except those for the trivial tangles. This suggests that we might
II.3. GLUEING VIA BORDERED SUTURED FLOER THEORY 51
always be able to find a parametrisation of the glueing surface such that the type D side is
bounded, simply by introducing Dehn twists of the tangles ends at the glueing surface.
Proof. The first two identifications follow directly from a comparison of the Heegaard
diagrams involved and the discussion following definition 3.6. In particular, each tensor
factor on the left of the second identification corresponds to an αβ-end, since the β-circle
at such an end intersects the silly α-arc twice. The difference between the contributions of
those intersection points to the Alexander and homological gradings are straightforward to
compute. For the third relation, we argue similarly, using
SFC(X ′T ) = ĈFT(T, s)⊗ V ⊗(l−c).
This identity follows from a small adaptation of the arguments of the proof of proposition 1.7:
Consider the surface obtained from ∂X ′T by deleting the component of R− that is contained
in ∂B3. Take an open collar neighbourhood N of this surface in X ′T , so we can write X
′
T
as a union of the closure of N and the complement of N in X ′T along their common bound-
ary. We can now apply the surface decomposition formula [Juh06b, proposition 8.6]. It is
straightforward to see that the sutured Floer homology of N is equal to V ⊗(l−c).
The final statement follows from Zarev’s general pairing theorem in bordered sutured the-
ory [Zar09, theorem 7.16]. 
Example 3.9 (glueing two 4-ended tangles). Suppose T is a 2-ended tangle and we cut it
into two 4-ended tangles T1 and T2. The glueing surface is a 3-punctured disc parametrised
by five arcs, of which three are silly arcs and the remaining two correspond to two adjacent
sites, say s1 and s2. Then the proposition above tells us that the generators of the type D
module B̂SD(T2) are in one-to-one correspondence with the generators of
ĈFT(T2, {s1})⊕ ĈFT(T2, {s2})
and the honest differentials in the type D structure correspond to the differentials in the
tangle Floer complexes. The statement for the generators is also true on the type A side
for T1, except that we obtain additional tensor factors. Then, after glueing, we get
ĤFT(T, ∅)⊗ V ⊗n = ĤFL(L)⊗ V ⊗n,
where n = 2 or n = 3, depending on whether glueing closes a component or not.
Proposition 3.10. Let Tp/q be the positive rational tangle corresponding to the frac-
tion pq with p > q ≥ 0 and gcd(p, q) = 1. Let XTp/q be the bordered sutured manifold with
the parametrisation specified by figure 23a. We label the two arcs which specify two sites of
the tangle by p and q.
Then the type D structure for Tp/q has the form of a loop, which corresponds to the single
β-curve in the Heegaard diagram in figure 23b in the following sense: The generators are
given by intersection points of the β-curve and the α-arcs. Each generator of the type D
structure has exactly two incoming or outgoing arrows which correspond to paths on the β-
curve from the corresponding intersection point to its two neighbours on the β-curve. The
algebra elements picked up by the differentials correspond to those paths on the arc diagram
after pulling the β-curve tight.
52 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
p
qTp/q
(a) The parametrisation of
the boundary of the tangle
complement XTp/q
q (p+ q) p
q
(p− q)
|
P1
|
Pq |
P ′p−q |
P ′1
|
Qq |Q1
τ1
τ2 ρ1
ρ2
ρ3
ε1
ε2
(b) A genus 0 Heegaard diagram for XTp/q . The blue bands repre-
sent parallel β-curve segments, the number of which is specified by
the label of the corresponding double arrow.
Figure 23. An illustration for proposition 3.10.
More explicitly, B̂SD(XTp/q) is obtained by glueing the following elementary “puzzle pieces”
together in such a way that the integers at the puzzle ends match:
p+ i p+ 1− i1 ≤ i ≤ q: Pi
ρ13τ12 ρ13
q + i q + 1− i1 ≤ i ≤ q: Qi
ε12
i+ 2q i1 ≤ i ≤ p− q: P ′i
ρ2ε12 ρ13
Our conventions for the algebra are explained in the proof, see also remark 3.12. If q > p ≥ 0,
the same holds, but for an explicit description in terms of such elementary pieces, we need
to replace P ′i by
i+ 2p i1 ≤ i ≤ q − p: Q′i
ρ13τ12ρ2
and to let the indices of the pieces Pi and Qi run from 1 to p.
Remark 3.11. The “puzzle piece” notation above is inspired by Hanselman and Watson’s
notation for their loop calculus in [HW15].
Remark 3.12. In the following proof, and also in all other computations of bordered
sutured invariants in this thesis, we actually compute the invariants of the mirrors of the
underlying bordered sutured manifolds. This is due to a late change of conventions in an
attempt to make them intrinsically consistent (we always use the right-hand rule to determine
orientations) and to match those most commonly used in the literature, see remarks 2.2
and 2.20 and figure 18. It is quite natural to draw Heegaard diagrams of tangles such that
the normal vector field points into the plane, see remark 2.2 and figure 16b. This is opposite
to standard conventions. However, the effect of reversing the orientation of the Heegaard
surface is marginal. It simply corresponds to a reversal of all arrows and gradings as well as
a reversal of the glueing algebra.
II.3. GLUEING VIA BORDERED SUTURED FLOER THEORY 53
|
c 1
|
c q
|
C
q
|
C
1
|
a
1
|
a
q
|
A
1
|
A
q
|f
1
|f
p
+
q
|F
p
+
q
|F
1
|d
1
|d
p
−q
|D
p
−q
|D
1
|l
1
|l
p
|L
p
|L
1
|g
1
|g
p
|G
p
|G
1
|
b 1
| b q
|
B
q
| B
1
|
P
1
| P
q
|
P
′ p−
q
| P
′ 1
|
E
1
| E
p
−q|
e p
−q
| e
1
|
Q
q
| Q
1
|
h
1
| h
p
|
H
p
| H
1
τ 1
τ 2
ρ
1
ρ
2
ρ
3
ε 1
ε 2
q
q
q
p (p
−
q)
(p
−
q)
q
p
F
ig
u
r
e
24
.
A
ni
ce
ifi
ed
H
ee
ga
ar
d
di
ag
ra
m
fo
r
th
e
p q
-r
at
io
na
lt
an
gl
e
54 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
Proof of proposition 3.10. We can easily turn the Heegaard diagram from fig-
ure 23b into a nice diagram, see figure 24. As noted in the previous remark, we now assume
that the normal vector field points out of the plane, so we actually compute the invariant
for the mirror image of XTp/q .
Since there is only one β-curve, the only domains that contribute to the type D module
are bigons with no or exactly one e-puncture, so we can compute the type D module very
easily from this nice diagram, using [Zar09, theorem 7.14]. The labelling of the additional
generators is such that there is always an identity morphism from a capital letter to the
corresponding lower-case letter with the same index.
fp+i Bi Ci
Ai bi ci
ai Pi fp+1−i
ρ1 τ1
1 1
τ2
1
τ1
ρ1τ1
(a) 1 ≤ i ≤ q
Fq+1−i Qi Li
Gi Hi li
gi hi Fi+q
ρ3 ε2
1
ε2
1
ε1
1
ε2 ρ3
(b) 1 ≤ i ≤ q
Di Ei P
′
i fi
di ei Li+q
Gi+q Hi+q li+q
gi+q hi+q Fi+2q
ρ3
ρ23
1
ρ2
ρ12
1
ρ1
ρ3 ε2
1
ε2
1
ε1
1
ε2 ρ3
(c) 1 ≤ i ≤ p− q
Figure 25. The three basic components of the type D module for Tp/q
The regions near the sutures are labelled by Greek letters with indices. We also use them
synonymously for the corresponding algebra elements. The arc diagram for the glueing
surface agrees with that of figure 26c. There are also two algebra elements ρ12 and ρ23 which
are picked up by rectangles with boundary regions (ρ1 and ρ2) and (ρ2 and ρ3), respectively.
These, together with the idempotents are all generators of the algebra that appear in the
type D module. An illustration of these is shown in figure 26d, where the idempotents are
represented by vertices and the algebra elements with moving strands by arrows connecting
the corresponding starting and ending idempotents. However, the algebra itself is slightly
larger, e. g., it contains an algebra element with a single moving strand from 8 to 10 whose
differential is non-zero. For details, see [Zar09, definitions 2.5 and 2.6].
Our conventions about the order for the algebra multiplication are such that ρ2ρ1 has two
moving strands, see also figure 54b. Note that τ2τ1 = 0 and similarly ε2ε1 = 0. Finally,
τ12 := τ1τ2, ε12 := ε1ε2 and ρ13 := ρ1ρ3.
II.3. GLUEING VIA BORDERED SUTURED FLOER THEORY 55
Let us now calculate the type D structure. For 1 ≤ i ≤ p + q, we have bigons from Fi
to fi, which contribute honest differentials Fi → fi. If we ignore these for a moment, the
(graph of the) type D module breaks up into the connected components shown in figure 25.
By cancelling the identity morphisms (using lemma A.19), we can simplify the diagrams
considerably and get the following:
(a) 1 ≤ i ≤ q : fp+i Pi fp+1−i,
(b) 1 ≤ i ≤ q : Fq+i Qi Fq+1−i,
(c) 1 ≤ i ≤ p− q : Fi+2q P ′i fi,
ρ1τ1τ2ρ1
ρ3ε1ε2 ρ3
ρ3ε1ε2ρ2 ρ1
noting that ρ3ε1ε2ρ12 = 0. The complete type D module can be obtained by connecting
these pieces and cancelling all Fi and fi. From this, it is apparent that the type D module
is of the form described in the proposition, again, noting that we have actually computed
the invariant of the mirror of XTp/q .
Suppose q > p ≥ 0. Then the corresponding Heegaard diagram is obtained by reflecting
the Heegaard diagram from figure 24 along a vertical line. This reverses the orientation of
the diagram again. Hence the invariant is obtained from the rules above by relabelling the
algebra (τ1 ↔ ε1, τ2 ↔ ε2, ρ1 ↔ ρ3) and the generators (Pi ↔ Qi, P ′i ↔ Q′i). The pieces Pi
and Qi turn out to be identical to the ones in the proposition, but the index i runs from 1
to p. Instead of the pieces P ′i , we get Q
′
i. 
Remark 3.13. XTp/q is bounded, except when (p = 1 and q = 0) or (p = 0 and q = 1).
Theorem 3.14. Two knots or links that are related by mutation of the (2,−3)-pretzel
tangle, oriented as in figure 26a have the same bigraded knot or link Floer homology, provided
that the two open strands p and q of the (2,−3)-pretzel tangle have the same colour. If the
orientation of one of the two strands is reversed, then in general this statement only holds
for δ-graded knot or link Floer homology.
Proof. We prove this result by calculating the type D invariant for the (2,−3)-pretzel
tangle with two different parametrisations. This computation is essentially the same as for
rational tangles – except that the nice diagrams are quite large: In one case, there are
over 400 generators, in the other nearly 3000. Therefore, we use the Mathematica pack-
age [BSFH.m] to compute the invariants, see notebooks [nb1] and [nb2]. The program first
computes the type D modules over a given strands algebra from nice diagrams. Then, it
cancels generators in a certain order, namely such that the generators from the non-niceified
Heegaard diagrams in Figures 26b and 26f survive. For more details, see appendix D, where
we explain how to use the package [BSFH.m]. Also note that in both calculations, we have
performed some homotopies to simplify the results, using the clean-up lemma A.21; for de-
tails, see the two notebooks.
Figure 27 shows the results of these computations (after reversing arrows, gradings and the
algebras, see remark 3.12): In both cases, we have arranged the generators in a grid ac-
cording to their Alexander bigrading, where  marks the origin (0, 0). The first Alexander
grading (corresponding to the p-strand) increases from top to bottom and the second grad-
ing (corresponding to the q-strand) increases from left to right. Also, the δ-grading of each
56 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
δ τ
ρε
qq
pp
a
b
c
d
(a)
T
he
first
param
etrisation,...
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
c
3
c
2 c1
d ′ d
y
3
y
2
y
1
x
1
x
2
y
3
y
2
y
1
x
1
x
2
ε
2
ε
1
ρ
3 ρ
2
ρ
1
τ
2 τ
1
(b)
...its
H
eegaard
diagram
,...
1
0987654321
ε
2
ε
1
ρ
3
ρ
2
ρ
1
τ
2
τ
1
(c)
...arc
diagram
...
•
• c
•
•d
•
•
ε
1
ε
2
ρ
3
ρ
1
2
ρ
2
3
ρ
1
ρ
2
τ
1 τ2
(d)
...and
algebra
δ τ
ρε
qq
pp
a
b
c
d
(e)
T
he
second
param
etrisation,...
b
b
b
b
b
b
b
b
b
b
b
b
b
b
a
2
a
1
d ′ d
y
3
y
2
y
1
x
1
x
2
y
3
y
2
y
1
x
1
x
2
δ
1
δ
2
τ
3
τ
2
τ
1
ρ
1
ρ
2
(f)
...its
H
eegaard
diagram
,...
1
0987654321
δ
2
δ
1
τ
3
τ
2
τ
1
ρ
2
ρ
1
(g)
...arc
diagram
...
•
• a
•
•d
•
•
δ
1
δ
2
τ
3
τ
1
2
τ
2
3
τ
1
τ
2
ρ
1 ρ2
(h)
...and
algebra
F
ig
u
r
e
26.
T
he
first
(a-d)
and
second
(e-h)
param
etrisation
for
the
(2,−
3)-pretzeltangle.
T
he
H
eegaard
diagram
s,arc
diagram
s
and
algebras
are
actually
the
ones
for
the
m
irror
of
the
tangle,i.e.the
(−
2,3)-pretzeltangle,see
rem
ark
3.12.
II.3. GLUEING VIA BORDERED SUTURED FLOER THEORY 57
δ−2x2d′ δ−
3
2x2c1 δ
− 3
2x2c2 δ
− 3
2x2c3

δ−
3
2x1c1 δ
−1dy2 δ−
3
2x1c2 δ
−1dy3 δ−
3
2x1c3
ρ2
τ12ρ2
ρ13τ12
ρ13
ρ13τ12ρ2ε12 ρ13τ12
ρ13
(a) The result from [nb1] for the first parametrisation
δ−2x2d′
δ−
3
2a1y1 δ
− 3
2a1y2 δ
− 3
2a2y1  δ− 32a1y3 δ− 32a2y2 δ− 32a2y3
δ−1dy2 δ−1dy3
τ2
ρ12τ2
τ13ρ12τ2δ12
τ13
τ13ρ12
τ13
τ13ρ12
(b) The result from [nb2] for the second parametrisation
Figure 27. Two type D modules for the (2,−3)-pretzel tangle from theorem 3.14.
Note that we have reversed all arrows, gradings and algebras since the notebooks
compute the invariants of the mirrors, see remark 3.12.
generator is shown in the exponent of δ in front of each generator, compare with exam-
ple 2.26. Furthermore, each generator is coloured corresponding to its site, see figures 26a
and 26e. The labelled arrows connecting these generators show the differential of the type D
structures and the algebra elements picked up along those differentials. Here, we use the
same conventions about the algebra as in the previous proposition.
Now, mutation about the y-axis corresponds to interchanging ρ↔ τ and ε↔ δ for algebra
elements and the sites a and c. Let us ignore the Alexander grading for a moment and just
consider the directed graphs consisting of the vertices and the solid arrows. Both of them
have four connected components: Two solitary generators, a loop of three generators and a
loop with four generators. We can choose an isomorphism between them that interchanges
generators of the sites a and c, preserves the δ-grading and changes the labelling of the arrows
exactly as specified by mutation about the y-axis. In other words, the δ-graded invariants
are homotopic to each other.
Let us now consider the Alexander grading. Obviously, the bigraded invariants are different,
already on the level of generators. This is to be expected, since the multivariate Alexander
polynomial is only mutation invariant if the colours of the two open stands in the mutating
tangle are the same, see theorem I.3.6. So we want to collapse Alexander gradings, i. e. add
the two Alexander gradings to a single Z-grading. This means that those generators on the
diagonals going from left to right and from bottom to top live in the same Alexander grading.
It is easy to see that the identification above can be chosen such that it also preserves this
single-variate Alexander grading.
Together with theorem 3.7 and the observation that mutation about the x-axis does not
change the knot or link, this finishes the proof of theorem 3.14. 
58 II. A HEEGAARD FLOER HOMOLOGY FOR TANGLES
Example 3.15. Let us reverse the orientation of a single strand, say the p-strand, and
see why bigraded mutation invariance fails. For example, this is the case for the Conway and
Kinoshita-Terasaka knots used in the counterexample from [OS03b, theorems 1.1 and 1.2],
see figure 28.
Changing the orientation of the p-strand has the effect of reversing the Alexander grading
p↔ p−1. In the two type D structures of the proof above, this means that the first Alexander
grading now increases from bottom to top. If we collapse the Alexander bigrading to a single
Z-grading, generators on the diagonals going from left to right and top to bottom now live
in the same Alexander grading. Obviously, these two invariants are not identical.
b
b
b
b
b
b
b
b
b
b
b
b
b
(a) The Kinoshita-Terasaka knot KT2,1
b
b
b
b
b
b
b
b
b
b
b
b
b
b
(b) C2,1, the Conway mutant of KT2,1
Figure 28. Ozsváth and Szabó’s counterexample from [OS03b] for bigraded muta-
tion invariance of ĤFK
Example 3.16. As a non-trivial example to which the proposition can be applied to
give the same bigraded link Floer homologies, we can consider the mutant pair in figure 29.
Indeed, their link Floer homologies both live in two consecutive δ-gradings and in each of
them, the Poincaré polynomial is
t−6 + 3t−4 + 3t−2 + 3t±0 + 3t+2 + 3t+4 + t+6.
See for example [OS06a, section 5.4] and [OS06b, section 3] for explicit calculations.
b
b
b
b
b
b
b
b
b
b
b
b
b
(a) The (3,−2, 2,−3)-pretzel link
a. k. a. Kinoshita-Terasaka link
b
b
b
b
b
b
b
b
b
b
b
b
(b) The (3,−2,−3, 2)-pretzel link
a. k. a. Conway link
Figure 29. An example for bigraded mutation invariance of ĤFL
CHAPTER III
Peculiar invariants for 4-ended tangles
In this chapter, we specialise to 4-ended tangles and reformulate the glueing structure
from section II.3 in terms of algebraic invariants which we call peculiar modules. In sec-
tion III.2, we prove a glueing formula for these invariants, based on a computer calculation
of a bordered sutured type AA bimodule, and give several applications thereof: For example,
in section III.3, we give new proofs of the oriented and the unoriented skein exact sequences
for link Floer homology, due to Ozsváth and Szabó [OS03a] and Manolescu [Man06]. We
also compute the peculiar module for the (2,−3)-pretzel tangle. It has certain symmetries
which, together with the glueing theorem, give a second proof of theorem II.3.14. In general,
we only obtain slightly weaker symmetry relations, see section III.4. Finally, in section III.5,
we explore the relationship between peculiar modules and the wrapped Fukaya category of
the 4-punctured sphere.
III.1. A glueing structure on ĤFT for 4-ended tangles
Definition 1.1. Let T be a 4-ended tangle. A peculiar Heegaard diagram for T is
obtained from a tangle Heegaard diagram for T by a local modification around the punctures,
as illustrated in figure 30: We collapse the four boundary components of Σ which meet the
α-arcs, thereby joining the four α-arcs to a single α-circle S1. Then we add a marked point
for each tangle end on either side of S1, pi on the front and qi on the back, and connect these
two points by an arc which intersects S1 exactly once and no other curve. We also contract
all other boundary components to points zj , so we get a closed Heegaard surface. We call
the union of the pi, qi and zj basepoints of the Heegaard diagram. Again, we need to
restrict ourselves to admissible diagrams, i. e. diagrams whose non-zero periodic domains
avoiding all basepoints have both negative and positive multiplicities.
i
back
front
(a) An original tangle end
qi
pi
(b) A new tangle end
Figure 30. The lazy way to count holomorphic curves near the boundary: Peculiar
Heegaard diagrams are obtained from those for tangles by local changes near the
boundary of the Heegaard surface.
59
60 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Remark 1.2. It is obvious that we can go from a peculiar Heegaard diagram back to
an ordinary tangle Heegaard diagram. The only reason for introducing peculiar Heegaard
diagrams is to avoid any bordered Heegaard Floer theory, so the proof of invariance of
the algebraic structures we are about to define is a minor adaptation of the one for link
Floer homology. In particular, note that the number of α-circles and β-circles in a peculiar
Heegaard diagram is the same.
Obviously, the Heegaard moves from lemma II.2.4 are equivalent to the following moves for
peculiar Heegaard diagrams:
• isotopies of the α- and β-curves away from the marked points and the arcs connect-
ing these,
• handleslides of β-curves over β-curves and handleslides of α-curves over α-curves
other than S1, and
• stabilisation.
Remark 1.3. The attribute “peculiar” should be considered as a homophone of “p-q-
liar”, a reference to the labels pi and qi we have chosen for the marked points, which is the
notation used in [AAEKO] in the context of the wrapped Fukaya category of the 4-punctured
sphere, see section III.5.
In the following, we add differentials to ĈFT that count the number of holomorphic
curves that have possibly non-zero multiplicities at the pi or qi. Unfortunately, the resulting
complex does not satisfy the relation ∂2 = 0 any more. However, it satisfies a slightly
modified ∂2-relation which enables us to promote ĈFT to a more sophisticated homological
invariant which we call a curved type D module, see example A.9. Let us recall this definition
here in slightly more down-to-earth terms.
Definition 1.4. Let I be a ring of idempotents and A a Z-graded algebra over I. Also
fix a central element ac ∈ Z(A) of degree −2. A curved type D structure over A is
a Z-graded I-module M together with an I-module homomorphism δ : M → A ⊗I M of
degree −1 satisfying
(µ⊗ 1M ) ◦ (1A ⊗ δ) ◦ δ = ac ⊗ 1M ,
where µ denotes composition in A. We call ac the curvature of M . A morphism be-
tween two curved type D structures (M, δM ) and (N, δN ) is an I-module homomorphism
M → A⊗I N . For two such morphisms f and g, their composition is defined as
(g ◦ f) = (µ2 ⊗ 1M ) ◦ (1A ⊗ g) ◦ f.
We endow the space of morphisms Mor(M,N) with a differential D defined by
D(f) = δN ◦ f + f ◦ δM .
Then indeed D2 = 0, since we have chosen ac to be central. This gives us an enriched
category over Com, the category of ordinary chain complexes over F2. The underlying
ordinary category is obtained by restricting the morphism spaces to degree 0 elements in the
kernel of D, giving us the usual notions of chain homotopy and homotopy equivalence, see
definition A.2 and example A.3.
III.1. A GLUEING STRUCTURE ON ĤFT FOR 4-ENDED TANGLES 61
Remark 1.5. It is interesting to compare curved type D structures to matrix factori-
sations as studied by Khovanov-Rozansky [KR04]. Given an algebra A over some field k, a
matrix factorisation of a potential w ∈ A consists of two free A-modules M0 and M1 with
two maps
(8) M0 M1
d0
d1
such that d1d0 = w. idM0 and d0d1 = w. idM1 . If M0 and M1 denote the k-vector spaces
generated by an A-basis of M0 and M1, respectively, we can regard d0 and d1 as maps
d0 : M0 → A⊗I M1 and d1 : M1 → A⊗I M0.
Then (M0⊕M1, d0 +d1) defines a curved type D structure over the k-algebra A. In general,
we cannot go in the other direction. For example curved complexes associated to manifolds
with torus boundary do, in general, not admit a splitting of the form (8). This is for the
simple reason that the total number of generators can be odd, see for example [HRW16, p. 9
fig. 6]. However, for peculiar modules, such splittings exist, see remark 1.9.
Definition 1.6. Let A∂ be the path algebra of the quiver
4•
1 • • 3
•
2
q4
p4
q1
p1
q3
p3
q2
p2
with relations piqi = 0 = qipi and I∂ the corresponding ring of idempotents. For an
illustration, see figure 31. We call A∂ the peculiar algebra.
More explicitly, A∂ can be described as follows. For i = 1, 2, 3, 4, let Ii = F2[ιi]/(ι2i = ιi) and
I∂ = I1× I2× I3× I4 the ring of idempotents, with one idempotent for each of the four sites
a, b, c and d of a 4-ended tangle. Let A∂ be the following algebra: As an additive group, it
is generated by closed oriented intervals on R with integer boundaries modulo overall shifts
of multiples of 4, i.e.
[t0, t1] ∼ [t0 + 4n, t1 + 4n] for all t0, t1, n ∈ Z.
Given two algebra elements a = [t0, t1] and a′ = [t′0, t′1], we define
a · a′ =
[t0, t1 + t′1 − t′0] if (t1 ≡ t′0 mod 4) and (t1 − t0)(t′1 − t′0) ≥ 00 otherwise.
The second condition in the first case means that the composition of two oppositely oriented
intervals is always zero. Furthermore, we define a bimodule action of I∂ on A∂ by
ιi[t0, t1]ιj =
[t0, t1] if (t0 ≡ i mod 4) and (t1 ≡ j mod 4)0 otherwise.
Thus, [t, t] = ιt for t ≡ t ∈ {1, 2, 3, 4} mod 4. For convenience, we usually write the elements
[t0, t1] with t0 6= t1 as q(t0+1)···(t1−1)t1 or pt0(t0−1)···(t1+1), depending on the orientation of the
interval, where we take the indices modulo 4 with an offset of 1. For example, we write
62 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
a
b
c
d
ι1
ι2
ι3
ι4
1
2 3
4
p1
q1
p2
q2
p3
q3
p4
q4
Figure 31. An illustration of the algebra A∂ . An algebra element can “walk” along
the front (the pis) or back (the qis) as far as it likes, but not around a puncture (the
pis and qis).
q41 for [3, 5] and p4 for [0,−1]. Furthermore, to simplify notation, we set
p = p1 + p2 + p3 + p4 and q = q1 + q2 + q3 + q4,
so we can write for example p4 = p4p3p2p1 + p3p2p1p4 + p2p1p4p3 + p1p4p3p2.
Definition 1.7. We can define a δ-grading on A∂ given by half the absolute length of
the interval:
δ([t0, t1]) :=
1
2 |t1 − t0|.
We can also define an Alexander grading A on A. However, some additional choices
are necessary for this. An oriented 4-ended tangle induces an orientation (i. e. labelling by
“in”/“out”) and colouring of the four punctures in figure 31. Note that the numbers of labels
of the same colour labelled “in” and “out” agree. Then for any such choice, we obtain an
Alexander grading on A as follows: Let |pi| = |qi| = a±1, where a is the colour of the ith
puncture, and choose “+” iff the ith puncture is labelled “in”. Then we can extend this
grading to the whole algebra such that |a · a′| = |a| · |a′| unless a · a′ = 0.
We sometimes denote the Alexander grading on generators by a superscript list of integers
(or half-integers), like a+1 for the single-variate or a(
3
2
,− 1
2
) for the multivariate Alexander
grading. Again, we can define a reduced Alexander grading Ar by identifying both colours
and a homological grading h as the difference
1
2A
r − δ.
Definition 1.8. Let pqMod be the category of curved complexes over A∂ with curvature
ac := p
4 + q4. We call the objects of this category peculiar modules.
Remark 1.9. Any homologically graded peculiar module (M,∂) admits a splitting like
a matrix factorisation (see remark 1.5), i. e. a splitting of M into two summands M0 and M1
such that ∂|M0→M0 and ∂|M1→M1 vanish. To see this, let us write M =
⊕
ιi.M and pick a
basis for each direct summand, so we get a basis forM . Consider two sequences of arrows in
∂ between two elements x and y of this basis. Suppose the labels of the arrows are given by
basic algebra elements a1, . . . , an and b1, . . . , bm, respectively. Since the homological grading
III.1. A GLUEING STRUCTURE ON ĤFT FOR 4-ENDED TANGLES 63
decreases along each arrow by 1, we have
h(x)− n =
n∑
i=1
h(ai) + h(y) and h(x)−m =
m∑
j=1
h(bj) + h(y),
so
(9) m− n =
n∑
i=1
h(ai)−
m∑
j=1
h(bj).
We want to show that this difference is even. The crucial observation is that the homological
grading difference between any two algebra elements in ιi.A∂ .ιj for fixed i, j ∈ {1, 2, 3, 4}
is an even integer. Thus, in the two sequences (a1, . . . , an) and (b1, . . . , bm), we can replace
any algebra element which is a product of qis by a product of pis without changing the
value of (9) modulo 2. But since the homological grading is compatible with the algebra
multiplication, (9) is equal modulo 2 to
h(a1 · · · an)− h(b1 · · · bm).
Now we can use the same argument again to see that this difference is even, hence so is
m− n. So we can split each connected component of the graph of (M,∂) by picking a base
vertex and taking the union of all vertices of even and respectively odd distance to the base
vertex.
Definition 1.10. Following the notation from chapter II, in particular definition II.2.7,
let φ ∈ pi2(x,y) be a domain between two generators x and y in T. Let nx(φ) denote the
multiplicity of φ at a basepoint x = pi, qi, zj . Let pi+2 (x,y) denote the subset of domains in
pi2(x,y) for which nx(φ) ≥ 0 for x = pi, qi and nzj (φ) = 0 for all j. Furthermore, set
np(φ) =
∑
i=1,2,3,4
npi(φ) and nq(φ) =
∑
i=1,2,3,4
nqi(φ).
Definition 1.11. Given a 4-ended tangle T and an (admissible) peculiar Heegaard
diagram for T , let CFT∂(T ) be the bigraded F2-module freely generated by elements in
T(T ), just as for ĈFT, see definition II.2.21. We can turn it into a left I∂-module as follows:
ιi.x =
x if s(x) = i0 otherwise for x ∈ T.
We define an endomorphism ∂ on CFT∂(T ) by
(10) ∂x =
∑
y∈T
∑
φ∈pi+2 (x,y)
µ(φ)=1
#M̂(φ) · ιs(x).pnp(φ)qnq(φ).ιs(y) ⊗I∂ y.
As usual, #M̂(φ) denotes the modulo 2 count of holomorphic curves in the moduli space
M̂(φ) associated with φ. Note that we choose the Alexander grading on A∂ that is induced
by T . We call (CFT∂(T ), ∂) the peculiar module of T .
Theorem 1.12. CFT∂(T ) is indeed a well-defined peculiar module. Furthermore, its
bigraded chain homotopy type is an invariant of the tangle T .
64 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Lemma 1.13. ∂ increases the δ-grading by 1 and preserves the Alexander grading.
(As usual, the grading on a tensor product is given by the sum of the gradings of the tensor
factors.)
Proof. Recall that the relative δ-grading on generators is defined via the Maslov index µ
of connecting domains, see definition II.2.15. When using peculiar Heegaard diagrams, we
replace punctures by basepoints. Of course, this does not change the Maslov index formula.
However, it does change the relationship between the Maslov index and the δ-grading. If
φ ∈ pi1(x,y) is a domain in a peculiar Heegaard diagram, let ϕ be the corresponding domain
in the usual tangle Heegaard diagram from section II.2. Then
δ(y)− δ(x) = µ(ϕ) = µ(φ)− 12#{pi, qi} −#{zj}.
The holomorphic discs contributing to the differential in (10) have Maslov index 1 and have
multiplicity 0 at the basepoints zj . So, the first claim follows from the definition of the δ-
grading on A∂ . The second statement follows directly from the definitions of the Alexander
gradings of generators and algebra elements. 
Lemma 1.14. For each x ∈ CFT∂, the sum on the right-hand side of (10) is finite.
Proof. The proof is essentially the same as in link Floer homology, see [OS05,
lemma 4.2]. Since CFT∂ is finitely generated, it is sufficient to show that the coefficient
of each y ∈ CFT∂ is a finite sum. Note that by the previous lemma, the difference of the
δ-gradings of x and y determines the δ-grading of a(φ). Thus, there are at most two choices
for the coefficients a(φ). So let us also fix the multiplicities of φ at the basepoints. We
can now argue as in the proof of [OS01, lemma 4.13], using admissibility of the underlying
Heegaard diagram. 
In the following, we need two analytical facts from [OS05].
Fact 1.15. [OS05, lemma 5.4] Given a homology class φ of β-injective boundary degener-
ations, write φ as a linear combination of connected components of Σrβ. Then its Maslov
index µ(φ) is equal to twice the sum of the coefficients. The same holds for α-injective
boundary degenerations. 
Fact 1.16. [OS05, theorem 5.5] Given a surface Σ of genus g, equipped with a set α of
(g + 1) attaching circles and a pseudo-holomorphic curve ψ of Maslov index µ(ψ) = 2 and
with non-negative multiplicities, then ψ is equal to one of the two connected components of
Σrα and the number of pseudo-holomorphic boundary degenerations in the homology class
of ψ is odd. 
Proof of theorem 1.11. Checking the ∂2-identity is analogous to the link case; we
can follow [OS05, proof of lemma 4.3] and count ends of moduli spaces of Maslov index 2
curves. We fix two generators x and z and consider the disjoint union of moduli spaces
M̂(φ), where φ varies over those curves in pi2(x, z) with µ(φ) = 2 and a(φ) = a for some
fixed a ∈ A∂ . (In particular, this fixes the multiplicities of φ at the pi and qi.) If there are
no boundary degenerations, there is an even number of ends, so the a⊗z-component of ∂2x
vanishes. If there are boundary degenerations, then by fact 1.15 above, they contribute at
III.1. A GLUEING STRUCTURE ON ĤFT FOR 4-ENDED TANGLES 65
least 2 to the Maslov index, so the remaining curve has to be constant, hence x = z. By
fact 1.16, we get a boundary degeneration for each component of Σ r α and Σ r β. But
since piqj = 0 for all i, j, β-injective boundary degenerations do not appear. α-injective
boundary degenerations do appear and they contribute exactly the terms p4 + q4. All other
ends appear in pairs again, so their contributions cancel.
It remains to show that the peculiar module is an invariant of the tangle T . Again, we adapt
the arguments for link Floer homology, more precisely the proofs of [OS05, theorems 4.4
and 4.7], which rely on the general machinery developed in [OS01]. We need to show that
the three Heegaard moves from remark 1.2 do not change the homotopy type of the peculiar
module of T .
• isotopies: Firstly, we need to study isotopies that do not change transversality of
α and β. In the case for closed manifolds, this is done in [OS01, theorem 6.1]. To
adapt the proof to our setting, we only need to replace the basepoint z by the marked
points pi and qi, and count multiplicities of holomorphic discs at those points in
the chain maps. Boundary degenerations do not appear because of fact 1.15. In
the case of closed components in T , admissibility ensures that all sums are finite.
Secondly, we need to check that exact Hamiltonian isotopies do not change the
homotopy type of CFT∂(T ). However, we can apply the same arguments as above
to adapt the proof for closed manifolds [OS01, theorem 7.3] to ours.
• handleslides: The main ingredient for proving handleslide invariance for Heegaard
Floer homology of closed manifolds and links therein are holomorphic triangles.
Only small modifications are necessary to adapt the proof from [OS05, section 6.3],
based on the arguments in [OS01, section 9] to our setting.
Let us consider a Heegaard diagram (Σ,α,β) and perform a handleslide of one
curve in β over another, away from all basepoints. Let γ denote the union of the
new β-curve with a small Hamiltonian translate of all unchanged β-curves. Also let
δ denote a small Hamiltonian translate of all β-curves. Then we consider the maps
Fαβγ : CFT
∂(α,β)⊗ CFT∂(β,γ)→ CFT∂(α,γ),
defined by using a holomorphic triangle count, recording multiplicities of the trian-
gles at the basepoints in the same way as in CFT∂ . Note that such triangle maps
preserve both Alexander and δ-gradings. Since the basepoint pi is in the same re-
gion as qi for all i, CFT∂(β,γ) = ĤF(β,γ, {pi = qi, zj}). By the proof of [OS05,
proposition 6.10], the latter is equal to Λ∗V , where V is some finite-dimensional
vector space, with a distinguished element Θβγ . Then Fαβγ induces a map
Ψαβγ : CFT
∂(α,β)→ CFT∂(α,γ),
by sending a generator x to Fαβγ(x⊗Θβγ).
In the same way, we can define the maps Ψαγδ and Ψαβδ. Associativity of the
triangle maps show that actually
Ψαγδ ◦Ψαβγ = Ψαβδ.
66 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Hence, it only remains to show that the map on the right is an isomorphism. For this
we adapt the arguments from [OS01, proof of proposition 9.8]. Let us consider gen-
erators in a fixed Alexander and δ-grading and a fixed site. If this set is non-empty,
let us fix a generator x0 in this set. Then, we can define a filtration as follows.
Choose an area function on Σ such that periodic domains avoiding basepoints have
signed area 0. Then, we define F(x) := −Area(φ), where φ ∈ pi∂2 (x0,x). Note
that for generators in the same site and Alexander grading, there is always such a
domain φ by lemma II.2.14. By the choice of our area function, F is independent of
φ. Now we can use the arguments from [OS01, proof of proposition 9.8] to see that
the restriction of Ψαβδ to our fixed set of generators can be written as the sum of
the identity map and a part which strictly lowers the filtration level.
We can now put all filtrations together to obtain a filtration on all generators. In-
deed, note that in a fixed δ-grading, there are only arrows between generators of
the same Alexander grading and the same site. Furthermore, because the δ-grading
on A∂ is non-negative, there are no arrows in Ψαβδ along which the δ-grading in-
creases. Since there are only finitely many generators, we can argue inductively,
starting at the lowest supported filtration level to see that Ψαβδ is an isomorphism,
as in [OS01, lemma 9.10].
Next, we look at handleslides of α-curves. Let us consider a Heegaard diagram
(Σ,α,β) and perform a handleslide of one curve in α over another, away from all
basepoints; as before let γ denote the union of the new α-curve with a small Hamil-
tonian translate of all unchanged α-curves. Also, let δ denote a small Hamiltonian
translate of all α-curves. Then we consider the maps
Fγαβ : CFT
∂(γ,α)⊗ CFT∂(α,β)→ CFT∂(γ,β),
as above. Since all pi lie in the same component of Σr(α∪γ), and similarly the qi,
CFT∂(γ,α)/(pi = 0 = qi) = ĤF(Σ,γ,α, {p1, q1, zj}) = Λ∗V as above. So we can
restrict this chain map in the second component to the unique element in highest
homological degree in Λ∗V , i. e. lowest in δ-grading. Together with the algebra
grading, this means that this restriction is still a chain map. Now we can argue as
above.
• stabilisation: For closed 3-manifolds, this is established in [OS01, section 10]. The
main ingredient is a glueing theorem for holomorphic discs which identifies the
moduli spaces before and after the glueing. Thus, stabilisation not only leaves the
generators unchanged, but also the structure maps, so the arguments carry over to
our case without any alteration. 
Example 1.17 (rational tangles). Figure 32 shows the peculiar modules of some very
simple 4-ended tangles. As shown in example II.2.5, every rational tangle T has a tangle
Heegaard diagram with just a single β-curve. Thus, we only count bigons in the differential
of the peculiar invariant, and only those that do not occupy both pi and qj . By tightening
the β-curve, we can assume that there are no honest differentials in CFT∂(T ), i. e. that every
bigon covers some pi or qi. Then CFT∂(T ) is a loop, and its representative in the 4-punctured
sphere is exactly the β-curve in the Heegaard diagram; compare this to proposition II.3.10.
III.1. A GLUEING STRUCTURE ON ĤFT FOR 4-ENDED TANGLES 67
b
b
d
b
b
b
b
b
d
c
b
a
b
b
b
b
d
c
b
a
δ0b(0,0) δ0d(0,0)
p43+q12
p21+q34
δ0a(
1
2
,− 1
2
) δ
1
2d(
1
2
, 1
2
)
δ
1
2 b(−
1
2
,− 1
2
) δ0c(−
1
2
, 1
2
)
p432
q341
p1
q4
p3
q2
p214
q123
δ0a(
1
2
,− 1
2
) δ−
1
2d(−
1
2
,− 1
2
)
δ−
1
2 b(
1
2
, 1
2
) δ0c(−
1
2
, 1
2
)
q1
p2
q234
p321
q412
p143
q3
p4
Figure 32. Basic rational tangles, their Heegaard diagrams and peculiar modules.
The superscripts of the generators specify the Alexander grading.
Example 1.18 (the (2,−3)-pretzel tangle). Figure 33 shows the computation of
CFT∂(T ) for the pretzel tangle from example II.2.26 and theorem II.3.14. First, we only
compute bigons and squares. Those are the labelled arrows in figure 33c. But there are
also other contributing domains. Grading constraints tell us that we can only get additional
morphisms between those generators which are connected by the other arrows, but in prin-
ciple, those could point in both directions. However, in each case, the connecting domains
in one direction either have negative multiplicities or occupy both pis and qis, so we can
only get arrows in one direction. From this and the ∂2-relation, we can deduce that all solid
arrows contribute. There are only eight remaining arrows (the dotted ones) and they can
only appear in pairs. But it is easy to see that we can homotope those dotted arrows away
(using the clean-up lemma A.21 for curved type D modules), so in any case, the complex is
homotopic to the invariant consisting of the solid black arrows only. We can then apply the
cancellation lemma A.19. We obtain a complex in which every arrow is paired with another
one going in the opposite direction and every generator is connected along the arrows to
exactly two other generators. A schematic picture of this complex is shown in figure 34a,
where these arrow pairs have been replaced by single unoriented edges, such that we obtain
a collection of loops. This motivates definition 1.20 below. Together with a glueing formula
like the one in theorem 2.5, mutation invariance can now be seen as the rotational symmetry
of these loops. Moreover, if we take into account the bigrading, we recover theorem II.3.14.
In figures 34b-d, the loops have been transferred onto separate 4-punctured spheres in such
a way that the vertices lie on the four arcs that connect the punctures and the unoriented
edges lie on the front or back of the spheres depending on whether they correspond to arrow
pairs labelled by pis or qis. The meaning of these loops will be discussed briefly in the next
section, see remark 2.2, and in section III.5 – for the moment, they are just a convenient
way to see the rotational symmetry.
68 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
qq
pp a
b
c
d
(a)
T
he
(2
,−
3
)-pretzeltangle
b
b
b
b
b
b
d
x
1x
2
b
a
2
a
1
p
1
q
1
q
2
p
2
4
2
1
3
b
b
b
b
b
b
b
b
d ′
y
1
y
2
y
3
b ′
c
3 c
2
c
1
p
4 q
4
p
3
q
3
15
46
2
3
(b)
A
H
eegaard
diagram
for
the
tangle
on
the
left
δ −
2x
2 d ′
δ −
32x
2 c
1
δ −
1by
1
δ −
32x
2 c
2
δ −
1by
2
δ −
32x
2 c
3
δ −
2x
2 b ′
δ −
1by
3
δ −
32a
1 y
1
δ −
32a
1 y
2
δ −
32a
2 y
1

δ −
32a
1 y
3
δ −
32a
2 y
2
δ −
32a
2 y
3
δ −
1d
y
1
δ −
2x
1 d ′
δ −
32x
1 c
1
δ −
1d
y
2
δ −
32x
1 c
2
δ −
1d
y
3
δ −
32x
1 c
3
δ −
2x
1 b ′
q
2
3
4
(1
5
)
p
4
q
2
3
(1
5
)
q
2
q
2
(1
2
6
)
p
3
(2
6
)
q
2
3
(1
)
q
2
3
(1
3
4
5
)
p
2
1
4
(4
)
q
2
p
3
(2
)
p
2
1
4
(4
6
)
q
3
q
2 1(2
)
p
2
1
4
3
(4
6
)
q
1
(3
)
p
3
2
(2
3
4
6
)
q
4
1
(3
)
p
4
3
2
(1
2
2
4
6
6
)
p
3
2
(2
6
)
q
4
1
(1
2
3
6
)
q
4
1
(3
5
)
p
3
2
(2
)
q
3
4
1
(1
3
3
4
5
5
)
p
3
2
(2
3
4
5
)
p
2
(2
)
p
1
1
(3
)
q
1
2
3
4
(1
5
)
q
1
2
3
(1
5
)
p
4
p
1
q
4
(3
)
q
1
2
3
(1
)
p
1
4
(4
)
p
1
4
(1
2
4
6
)
p
1
q
4
(3
5
)
p
1
(3
4
5
)
q
3
p
1
4
(4
6
)
p
1
4
3
(4
6
)
(c)
T
he
peculiar
invariant
for
the
pretzeltangle
above
F
ig
u
r
e
33.
A
com
putation
of
a
peculiar
invariant
for
a
non-rationaltangle,see
exam
ple
1.18
III.1. A GLUEING STRUCTURE ON ĤFT FOR 4-ENDED TANGLES 69
⊙
b
−2
b
− 3
2
b
−1
b
− 3
2
b
−1
b
− 3
2
b
− 3
2 b− 3
2
b
− 3
2 b− 3
2
b
− 3
2 b− 3
2
b
− 3
2
b
−1
b
− 3
2
b
−1
b
− 3
2
b
−2
(a) A schematic picture of the result. Generators correspond to vertices, arranged according to
their Alexander grading and labelled by their δ-grading. The dotted edges correspond to pairs of
arrows labelled by pis, the solid ones to pairs of arrows labelled by qis.
b
b
bb
b
b
(b)
b
b
b
b
b
b
(c)
b
b
b b
b
b
(d)
Figure 34. The final result of the computation from example 1.18 and figure 33.
(b)–(d) show the three loops from (a) separately on 4-punctured spheres.
Proposition 1.19. For any 4-ended tangle T , CFT∂(T ) is homotopic to a peculiar
module without any identity components in the differential.
Proof. The identity is the only algebra element of A∂ with δ-grading 0 and the only
algebra elements with δ-grading 1 are pipi−1 and qiqi+1, so there cannot be any arrow from
a generator to itself. So we can repeatedly apply the cancellation lemma A.19. 
Definition 1.20. Let M be a peculiar module without any identity components. As
usual, we can think of M as a labelled directed graph whose vertices are the generators and
whose arrows correspond to morphisms, labelled by pis and qis. M is called a loop, if this
graph is connected and every vertex is 4-valent. Note that we can represent any such loop
as an immersed curve on the 4-punctured sphere in the same way as in the previous two
examples. In analogy to the torus boundary case from [HW15], we call M loop-type if M
is homotopic to a collection of loops.
Questions 1.21. Is CFT∂(T ) loop-type for every tangle T? Do homotopy classes of
loop-type peculiar modules correspond to homotopy classes of loops on the 4-punctured sphere?
If so, what does the number of components tell us about T?
70 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Remark 1.22. We do not expect every peculiar module to be loop-type. Consider the
following example:
b d
d b
p43+q12
p21
p21+q34
p43
p43+q12
p21+q34
In the Fukaya category setting, i. e. under the functor L from theorem 5.10, this corresponds
to the cone of the non-identity morphism between the loop for the trivial tangle from figure 32
and a Hamiltonian push-off, which is a prototypical example of a local system from [HKK14],
see remark 5.18.
In proposition 1.19, we have seen that one can eliminate all identity components in a
peculiar module. In view of questions 1.21, the following question might also be interesting.
Question 1.23. Is every peculiar module homotopic to one with p-, p2-, p3-, q-, q2- and
q3-components only? In other words, can we always eliminate all identity components and
“long” algebra elements?
Proposition 1.24. Let T be a 4-ended tangle and L the link obtained by closing T at
the sites a and c like so:
Ta c
Then
ĈFL(L)⊗ V i ∼= CFT∂(T )/(p1 = p2 = q3 = q4 = 1, q1 = q2 = p3 = p4 = 0)
∼= CFT∂(T )/(p1 = p2 = q3 = q4 = 0, q1 = q2 = p3 = p4 = 1),
where i = 0 or 1, depending on whether L has two or one more closed component(s) than T ,
respectively. For the other two opposite sites, we obtain a similar formula by a cyclic per-
mutation of the indices.
Proof. If we delete those basepoints in a peculiar Heegaard diagram of T that cor-
respond to the variables that we set equal to 1, we obtain a Heegaard diagram for L. In
ĈFL(L), we only count those holomorphic curves that stay away from the remaining base-
points, so we need to set those algebra elements equal to 0. 
Remark 1.25. One might hope to get the other flavours of link Floer homology from the
same kind of idea as in the proof of the previous proposition. However, this does not work
quite as well because of the relation piqi = 0. As soon as we do not set at least one of the
two marked points at each tangle end equal to 0, we would have to count more holomorphic
curves than we do for CFT∂(T ).
Suppose, the first two of questions 1.21 have a positive answer. Then, we might still hope to
reconstruct from CFT∂(T ) the other flavours of the link Floer homology of the closure of T
as follows: Consider the embedded loop(s) corresponding to CFT∂(T ) as a β-circle and the
four α-arcs as a single α-circle as before. Then we can calculate a new complex from this by
also counting those bigons which occupy both pi and qi.
III.2. PAIRING 4-ENDED TANGLES 71
We end this section with some simple observations about CFT∂ .
Observation 1.26. Let T ⊂ B3 be a 4-ended tangle and suppose its peculiar module is a
collection of loops. As in example 1.18/figure 34, we can transfer them onto the 4-punctured
sphere ∂B3 r ∂T . By considering the Alexander grading, we see that every loop is in the
kernel of
pi1(∂B
3 r ∂T )→ pi1(XT )→ H1(XT ),
where XT = B3 r ν(T ) is the complement of a tubular neighbourhood of the tangle T .
Observation 1.27. By definition, the Alexander grading corresponding to each closed
component vanishes on A∂ . Also, the differential of a peculiar module preserves the Alex-
ander grading by lemma 1.13. Thus, ĈFT(T ) decomposes into the direct sum over the
Alexander gradings of closed components.
Observation 1.28. In [OS02], Ozsváth and Szabó showed that the link Floer homology
of alternating links is completely determined by the Alexander polynomial (and, to be precise,
the signature, but this is only needed to fix the absolute grading). The proof generalises
immediately to ĤFT, using the generalised clock theorem I.1.17.
Question 1.29. Given an alternating tangle T , is the bigraded chain homotopy type of
CFT∂(T ) determined by ∇sT ?
III.2. Pairing 4-ended tangles
In this subsection, we prove a glueing formula for CFT∂ : Given the peculiar modules
of two 4-ended tangles T1 and T2, we compute the Heegaard Floer homology of the link L
obtained by glueing T1 to T2, up to at most three stabilisations. The proof is essentially
a calculation of a bordered sutured type AA bimodule, for which we use a computer since
the number of generators in the bimodule is rather large. As a warm-up, we prove a special
case of the glueing formula, namely for T2 being a trivial tangle, where we can do this
calculation by hand. This corresponds to computing ĤFL of the closure of a tangle. Of
course, we already know how to do this, see proposition 1.24. There are three reasons for
looking at this again. Firstly, it illustrates how the proof of the general glueing formula
works. Secondly, the result that we get in this special case is as nice as one might hope,
which, unfortunately, we cannot say about the general glueing formula in its present state,
see conjecture 2.7. Thirdly, a comparison of the two methods for computing ĤFL of the
closure might give us interesting symmetry relations on CFT∂ .
Theorem 2.1. Let T be a 4-ended tangle and L the link obtained by closing T at the
site a and c like so:
Ta c
For a site s, let C(s) be the strictly unital type A structure over A∂ defined by the labelled
graph shown in figure 35a. Then
ĈFL(L)⊗ V i ∼= CFT∂(T ) C(a) ∼= CFT∂(T ) C(c)
72 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
where i = 0 or 1, depending on whether L has two or one more closed component(s) than T ,
respectively. For the other two opposite sites, we obtain similar formulas by cyclic permuta-
tions of the indices.
Remark 2.2. If CFT∂ is loop-type, we can interpret this pairing theorem in terms of
Lagrangian intersection homology. Consider figure 35b. The blue lines denote a 1-skeleton R
with two 0-cells labelled p and q and four 1-cells a, b, c and d, each of them connecting p
and q, such that R is a deformation retract of the 4-punctured sphere. Given a collection of
loops representing a loop-type peculiar module of a tangle T , we write it as a cycle on R.
By shortening this cycle as much as possible, we get a unique one. Now consider the violet
loop C, which is a Hamiltonian (=area-preserving) push-off of a loop on R representing the
trivial tangle. Then we see four intersection points of C with R, which we label a1, a2,
b1 and b2. These correspond to the generators in C(a). The differentials in C(a) can also
be computed from this picture; they come from counting convex bigons that connect the
b1 b2
a1 a2
1+(p21,q12)
(q2)
(p1,q12)
(p1,q1)
(p21,q1)
(p2)
(a) The type A structure C(s) for closing a 4-ended tangle at the site s = a. The identity action
is implicit. We similarly define C(s) for any of the other three sites of a 4-ended tangle by cyclic
permutations of the sites and indices corresponding to rotations of figure 31.
4
32
1
p
1
2 3
4
q
q
q
q
4
32
1
a
b
c
d
b
a1
b
a2
b
b2
b
b1
(b) A combinatorial model for pairing any Lagrangian in the wrapped Fukaya category of the 4-
punctured sphere with the violet loop respresenting a trivial tangle. The boundary of the picture
is identified to a point. The blue lines denote a 1-skeleton on which the Lagrangian lives.
Figure 35. Closing a 4-ended tangles at the site s = a
III.2. PAIRING 4-ENDED TANGLES 73
generators and avoid all punctures. The labelling can be computed from the multiplicities
of the bigons in the regions near the 0-cells p and q. Our conventions are such that the
curve on R plays the role of a β-curve and C the one of an α-curve. Furthermore, the normal
vector field points into the plane, i. e. the sphere. For example, we see two bigons connecting
b2 to b1: One of them stays away from the 0-cells p and q, so it contributes 1, the other picks
up p2, p1, q1 and q2.
a
b
c
d
T
(a) The bordered sutured structure on XT
b
a
d
(b) The bordered sutured manifold C(a)
Figure 36. An illustration for the proof of theorem 2.1. Pairing these two bordered
sutured manifolds gives the sutured manifold of the link obtained by closing T at
the site a, plus an extra pair of meridional sutures, if the two tangle ends adjacent
to the arc a do not belong to the same component of T .
Proof of theorem 2.1. The proof is essentially a computation of a bordered sutured
type A structure corresponding to closing a tangle, namely the one for the bordered sutured
manifold C(a) shown in figure 36b. A Heegaard diagram for it is shown in figure 37a. We
only compute the full subcomplex of the corresponding type A structure consisting of those
generators which occupy exactly two arcs of {a, b, d}; this part is shown in figure 37b. Its
computation is more or less straightforward, noting that the multiplicities of all regions are
either 0 or 1. The only domain which could contribute and which is not a polygon is A+B.
To solve this problem, we can argue as follows: Pairing the type A structure with the type
D structure consisting of a single generator in idempotent {d} and no differential gives 0.
By symmetry of C(a), pairing with a single generator in idempotent {b} should give 0 as
well. However, if A + B did not contribute, the homology of the paired complex would be
2-dimensional.
We can simplify this type A structure by cancelling the identity map between the generators
b2y1z2 and b1x2z1 and then do some homotopies using lemma A.21 to obtain the following
complex, which we denote by C:
b1 b2
a1 a2
1+(δ2τ2,τ13δ13)
(δ13)
(τ2,τ13δ13)
(τ2,τ13)
(δ2τ2,τ13)
(δ2)
74 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
b
b
b1
b
y2
b
y1
b
b2
b
x1
b
x2
b
a
b
z1
b
z2
A
B
B
τ3τ2
τ1
δ3
δ2 δ1
b
a
d
(a) A Heegaard diagram for the bordered
sutured manifold in figure 36b. The normal
vector field points into the plane.
b2y1z2 b1x2z1
b1x1z1 b2y2z2
ay1z2 ay2z2
τ1‘2‘3
1(A)
τ1,2‘3(A)
1(AB)+δ1,2‘3τ1,2‘3(A)
δ13
τ1,2‘3δ13(A)
δ13δ2(A)
δ1,2‘3
τ1,2′3
τ1,2‘3δ2
δ2
δ2
(b) The type A structure computed from the
Heegaard diagram on the left
Figure 37. The computation of the type A structure for the bordered sutured
manifold in figure 36b. There are only two interior regions, labelled by A, B. If
a domain containing one or both of these contributes to the structure map, we
record it in brackets in the type A structure. A ‘ (,) in a subscript means that the
corresponding α-arc is (not) occupied. For more details, see [nb3].
Let A∂12 be the quotient of A∂ by ι3 and all elements pI and qI where I ∩ {3, 4} 6= ∅. Let
pi12 denote the quotient map. Note that A∂12 is generated by {p1, p2, p12, q1, q2, q12} as an
I∂ /ι3-module. Furthermore, if B denotes the dg algebra underlying C, we can define an
I∂ /ι3-algebra homomorphism
pi : B → A∂12
by sending τ2 7→ p1, δ2 7→ p2, δ2τ2 7→ p21, τ13 7→ q1, δ13 7→ q2, τ13δ13 7→ q12 and all other
basic algebra elements to 0. Note that this is a homomorphism of dg algebras, where we
define the differential on A∂12 to vanish. Hence, by remark A.16, pi induces a functor Fpi
from type A structures over B to those over A∂12, that respects chain homotopies. Note that
Fpi(C) = C(a).
Next, let D(T ) be the type D structure computed from a Heegaard diagram, where the silly
α-arcs do not intersect any β-curves. Then
SFC(XT ∪ C(a)) = D(T ) C = Fpi(D(T )) C(a).
By construction, XT ∪C(a) is equal to the link complements of L with some meridional su-
tures. If the two open components of T are not joined up, each link component of L carries a
single pair of meridional sutures, so SFC(XT ∪C(a)) agrees with ĈFL(L). Otherwise, there
is an extra pair of meridional sutures on the single new closed component, which contributes
the tensor factor V .
Since applying (−  C(a)) to Fpi12(CFT∂(T )) and CFT∂(T ) gives the same result, it only
remains to compare Fpi(D(T )) to Fpi12(CFT∂(T )). We claim that they are identical, pro-
vided the underlying Heegaard diagram for D(T ) is obtained from the one for CFT∂(T )
III.2. PAIRING 4-ENDED TANGLES 75
by removing the arc for site c and performing the obvious modifications near the tangle
ends. Then, the underlying generator sets in sites a and b are the same. Furthermore, the
moduli spaces for the algebra elements corresponding to the basic elements in A∂12 are the
same by the next lemma, since the underlying domains are obtained from one another by
adding/removing the silly α-arcs. 
Lemma 2.3. Let D and D′ be two domains which only differ in a small region of multi-
plicity 1 as follows:
D
←→ D′
Then, for a suitable choice of complex structure, D contributes iff D′ does.
Proof. This follows from the same arguments as [Han13, proposition 2.7]. 
Remark 2.4. In the proof above, we could alternatively have also added two meridional
sutures to the manifold, thereby making the Heegaard diagram planar and the computation
purely combinatorial. This comes at the expense of getting an extra tensor factor after the
pairing. However, this factor can be removed, since the type A structure is homotopic to a
direct sum of two identical copies of C(s). This computation is done in the notebook [nb3],
see also appendix D, in particular figure 53. For the general glueing formula, one might hope
something similar would happen. However, this appears not to be the case.
Theorem 2.5. Let T1 and T2 be two 4-ended tangles and L the link obtained by glueing
them together according to the following picture
T1 T2
Then there exists a strictly unital type AA structure P over A∂ such that
ĈFL(L)⊗ V i = CFT∂(T1) P  CFT∂(T2)
where i = |T1|+ |T2| − |L| ∈ {2, 3}.
Remark 2.6. The type AA structure P looks almost like the direct sum of four copies
of the complex Q shown in figure 38. However, between those four identical blocks, there
are structure maps which I do not seem to be able to get rid of. They also have a certain
symmetry. Schematically, P looks as follows:
Q Q
Q Q
g
f h
g
f
76 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
{q3q2, p2p3} ι4
ι2 {q1q4, p4p1}
( {q3q2, p2} ι4 ) ( {q3} ι4 )
( ι2 {q1} ) ( ι2 {q1q4, p4} )
ι3 {p1p2, q2q1}
{p3p4, q4q3} ι1
( ι3 {p1} ) ( ι3 {p1p2, q2} )
( {p3p4, q4} ι1 ) ( {p3} ι1 )
( {p3} ι4 )
( ι3 {p2, q2q1} )
( ι3 {p2, q2} )
( {q2, p2p3} ι4 )
( ι3 {q1} )
( {q2, p2} ι4 )
( ι2 {q4, p4p1} )
( {q3} ι1 )
( ι2 {q4, p4} )
( ι2 {p1} )
( {p4, q4q3} ι1 )
( {p4, q4} ι1 )
1:(ca) 2:(ca)
3:(bd) 4:(bd)
5:(ba) 6:(ba)
7:(cd) 8:(cd)
Figure 38. The complex Q, one of four identical components of the type AA
structure P from theorem 2.5. This is an output of the notebook [nb4], so by re-
mark II.3.12, we actually need to reverse all arrows and the algebra.
We will not define the maps f , g or h since they are not particularly enlightening; the
interested reader should look at the computation in [nb4] for details. The complex Q itself
does not look particularly simple either, meaning that I am at the moment unable to find an
interpretation of Q in terms of the Lagrangian intersection homology as in the case of closing
a tangle, see remark 2.2 and figure 35b. Instead, I had expected the type AA structure P ′
from figure 39, which does admit such an interpretation. Calculations for loop-type peculiar
modules suggest that we can also use P ′, giving rise to the following conjecture; also compare
this to conjecture 5.16.
Conjecture 2.7. Theorem 2.5 remains true, if we replace P by the type AA structure P ′
defined by the labelled graph in figure 39 and set i = |T1|+ |T2| − |L| − 2 ∈ {0, 1}.
Proof of theorem 2.5. We follow the same line of argument as in the proof of the-
orem 2.1, except that we let the computer do the main calculation, using the Mathematica
notebook [nb4]. We glue two bordered sutured manifolds associated with the tangles T1
and T2 together along a third bordered sutured manifold P , which is topologically just a
thickened 4-punctured sphere as illustrated in figure 40. The entire calculation of the bor-
dered sutured type AA structure for P , including cancellations and homotopies, are done
in [nb4] – most of them in some automated fashion.
III.2. PAIRING 4-ENDED TANGLES 77
AA dd
BA cd
BB cc
aa DD
ba CD
bb CC
(p432|q1)
(p32|q1)
(p2|−)
(−|−)
(p43|q1)
(p3|q1)
(p3|q41)
(p4|−)
(−|q4)(−|q2)
(p43|q12)
(p3|q12)
(p3|q412)
(−|−)
(p32|q41)
(−|−)
(q234|p1)
(q34|p1)
(q4|−)(q2|−)
(−|p2)
(q23|p1)
(q3|p1)
(q3|p21)
(q34|p21) (−|−)
(−|p4)
(q23|p14)
(q3|p14)
(q3|p214)
(a) The expected type AA structure P ′. The identity action is implicit. Part of the actual type AA
structure used in theorem 2.5 is shown in figure 38, see remark 2.6.
1
2 3
4
p
1
2 3
4
q
q
q
q
4
32
1
q
1
2 3
4
p
3
2 1
4
a
b
c
d
a
b
c
d
b
AA
b
aa
b
BB
b
bb
b
BA
b
ba
b
CC
b
cc
b
CD
b
cd
b
DD
b
dd
(b) The expected combinatorial model for pairing in the wrapped Fukaya category of the 4-
punctured sphere; compare this to figure 35b. Again, the boundary of the picture is identified to
a point. The blue curves denote a 1-skeleton and the violet ones a Hamiltonian translate thereof,
with the labelling p and q reversed.
Figure 39. The expected glueing structure in conjecture 2.7
78 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
a
b
c
d
T1
(a) The bordered sutured structure on XT1
a
b
c
d
T2
(b) The bordered sutured structure on XT2
a
b
c
d
a
b
c
d
(c) The bordered sutured manifold P
Figure 40. Pairing the bordered sutured manifold for T1 to the inside of P and the
one for T2 to the outside gives the sutured manifold of the link obtained by pairing T1
and T2 according to the picture in theorem 2.5, plus some extra pairs of meridional
sutures. There are three additional pairs of sutures if the four open components
of T1 and T2 glue up to a single closed component; otherwise, there are only two
superfluous meridional suture pairs.
III.2. PAIRING 4-ENDED TANGLES 79
Just as for closing a tangle, the main point of this proof is that we can homotope the type
AA structure for P , which is defined over some complicated bordered sutured algebras B1
and B2, to one where we can replace B1 and B2 by some suitable quotient of A∂ without
changing the result under any glueing. We quotient out by p1 and q1 for the first glueing
surface, i. e. the one for T1 on the “inside” of P in figure 40c. For the second glueing surface,
we quotient out by p3 and q3. Then, by using Heegaard diagrams for XT1 and XT2 , where
the silly arcs do not intersect any β-curves, the same argument as in the proof of theorem 2.1
tells us that
SFC(XT1 ∪ P ∪XT2) = CFT∂(T1) P  CFT∂(T2).
Now we just need to count the number of meridional sutures in XT1 ∪P ∪XT2 to determine
how often we need to stabilise the link Floer homology of L. 
Theorem 2.8. A 4-ended tangle T is rational iff CFT∂(T ) is homotopic to a single loop
that corresponds to an embedded loop on the 4-punctured sphere.
Proof. The only-if direction is simply a calculation that we did in example 1.17. The
opposite direction follows essentially from the fact that link Floer homology detects the knot
genus and a cut-and-paste argument.
Suppose CFT∂(T ) is a single loop which corresponds to an embedded loop on the 4-punctured
sphere. It divides the sphere into two disc components, each of which has at least one
puncture, since the loop is not nullhomotopic. By observation 1.26, there are exactly two
punctures in each disc, so CFT∂(T ) agrees with CFT∂(T ′) for some rational tangle T ′. Then
also CFT∂(m(T )) agrees with CFT∂(m(T ′)). Let L be the link obtained by pairing T with
its mirror m(T ). If we glue T ′ to m(T ′), we obtain the 2-component unlink ©q©, so by
theorem 2.5
ĤFL(L)⊗ V ⊗2 = CFT∂(T ) P  CFT∂(m(T ))
= CFT∂(T ′) P  CFT∂(m(T ′)) = ĤFL(©q©)⊗ V ⊗2.
We now apply the fact that link Floer homology detects the link genus [OS06a], so L is the
2-component unlink. The following lemma finishes the proof. 
Lemma 2.9. Let T1 and T2 be two 4-ended tangles that glue together to the 2-component
unlink. Then either T1 or T2 is a rational tangle.
Proof. Let S be the 4-punctured sphere along which we glue T1 and T2. Let U be
the sphere that separates the two unknot components and assume that S and U intersect
transversely in a disjoint union of circles. We now proceed by induction on the number of
circles in S ∩U . First of all, this intersection is non-empty, since U is separating. So we can
always find a curve γ that bounds a disc D in U . If γ also bounds a disc D′ in S, D ∪D′
bounds a 3-ball, which we can use as a homotopy for U to remove γ, so we are done by the
induction hypothesis. If γ does not bound a disc in S, it separates two punctures from the
other two. So D separates the two strands in T1 or T2. They must obviously be unknotted,
since the connected sum of two knots is the unknot iff both knots are unknots. Thus either
T1 or T2 is rational. 
80 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
III.3. Skein relations
We start with a slight generalisation of Ozsváth and Szabó’s exact triangle [OS03a] which
categorifies the oriented skein relation for the Alexander polynomial (lemma I.2.8). Note,
however, that we only get a twice stabilised version, due to the shortcomings of our glueing
theorem 2.5.
Theorem 3.1 (n-twist skein exact triangle). Let Tn be the positive n-twist tangle, T−n
the negative n-twist tangle and T0 the trivial tangle, see figure 41. Furthermore, let V be
a 2-dimensional vector space supported in degrees δ0tn and δ0t−n. Then there is an exact
triangle
δ−
n
2 CFT∂(Tn) δ
n
2 CFT∂(T−n)
CFT∂(T0)⊗ V
ϕn
ϕn preserves the (single-variate) Alexander grading and changes δ- and homological gradings
by +1 and −1, respectively; the other two maps preserve the all three gradings. Moreover,
given three links Ln, L−n and L0 in S3, which agree outside a closed 3-ball and in this closed
3-ball agree with the 4-ended tangles Tn, T−n and T0, respectively, then the above triangle
together with the glueing theorem induces an exact triangle
ĤFL(Ln)⊗ V ln ĤFL(L−n)⊗ V l−n
ĤFL(L0)⊗ V l0+1
where for i ∈ {n,−n, 0}, li is either 2 or 3, depending on whether the two strands in Ti
belong to different or the same components in Li, respectively.
Remark 3.2. Similar results hold for other orientations; for n even, also multivariate
Alexander gradings are preserved.
Proof. It is straightforward to compute CFT∂(Tn) and CFT∂(T−n) from genus 0
Heegaard diagrams, they are both shown in figure 42, the former on the left, the latter
on the right. If n is odd, the dashed lines denote a sequence of alternating generators in
sites a and c, connected by pairs of morphisms, labelled alternatingly by pis and qis. For
even n, the two components are connected by similar sequences along the dotted lines. The
horizontal arrows in figure 42 describe ϕn.
n
 ...
(a) Tn
n
 ...
(b) T−n (c) T0
Figure 41. The basic tangles for the skein exact sequence from theorem 3.1
III.3. SKEIN RELATIONS 81
By cancelling all identity components of the mapping cone of ϕn, we get two copies of
CFT∂(T0) = δ
0b0 δ0d0.
p43+q12
p21+q34
Thus, we can write CFT∂(T0)⊗ V as a cone of CFT∂(Tn) and CFT∂(T−n), which gives rise
to the exact triangle of the required form. 
Next, we give a new proof of a theorem by Manolescu [Man06, theorem 1]. Again,
we only get a twice stabilised version. Like Manolescu’s triangle, ours does not preserve
any gradings. Note that Manolescu probably uses slightly different conventions from ours,
because the two triangles only look the same after reversing the direction of the three arrows.
Theorem 3.3 (resolution skein exact triangle). There is an exact triangle
CFT∂( ) CFT∂( )
CFT∂( )
ϕ
Moreover, given three links L0, L1 and LX in S3, which agree outside a closed 3-ball and in
this closed 3-ball agree with the 4-ended tangles T0 = , T1 = and TX = , respectively,
then the above triangle, together with the glueing theorem induces an exact triangle
ĤFL(L0)⊗ V l0 ĤFL(L1)⊗ V l1
ĤFL(LX)⊗ V lX
where for i ∈ {0, 1, X}, li is either 2 or 3, depending on whether the two strands in Ti belong
to different or the same components in Li, respectively.
δ−
1
2 c1−n δ
1
2 c1−n
δ0b−n δ0d−n
δ−
1
2a1−n δ
1
2a1−n
δ−
1
2 cn−1 δ
1
2 cn−1
δ0dn δ0bn
δ−
1
2an−1 δ
1
2an−1
1
p21+q34
p3
p214
q2q341
p321
p4
q234q1
1
1
p1
p432
q4 q123
p43+q12
p143
p2
q412 q3
1
Figure 42. The morphism ϕn : δ−
n
2 CFT∂(Tn)→ δ n2 CFT∂(T−n)
82 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Proof. The map ϕ is given by (the horizontal arrows in) the following diagram on the
left:
b c
d a
p43+q12p21+q34
q3
p2
p14+q23p32+q41
q1
p4
∼=
b c
d a
p143
p2
q412
q3
p321
p4
q234
q1
Using lemma A.21, we see that it is homotopic to the diagram on the right, which is
CFT∂( ). Now apply the same arguments as in the proof of theorem 3.1. 
Proposition 3.4. There are two morphisms
δ−
1
2 t±1 CFT∂
( )
→ CFT∂
( )
whose mapping cones are homotopic to loop-type complexes, representing the following
“figure-8” loops:
b
b b
b
and
b
bb
b
By the symmetry of these loops, taking the mirror gives us these loops as the mapping cone
of maps from the negative crossing to the trivial tangle.
Proof. The two maps between the peculiar invariants of the two tangles look as follows:
δ−
1
2d+1 δ
1
2d+1
δ0a0
δ0c0
δ−
1
2 b+1 δ
1
2 b−1
1
q4q123
p432
p1
p2
q3
p43+q12
p21+q34
p3
p214
q2
q341 and
δ−
1
2d−1 δ
1
2d+1
δ0a0
δ0c0
δ−
1
2 b−1 δ
1
2 b−1
p4
q1
p4q123
p432
p1
1
p43+q12
p21+q34
p3
p214
q2
q341
We now cancel the identity arrows in both mapping cones and respectively get
δ0a0 δ−
1
2 b+1
δ
1
2 b−1 δ0c0
q341q2
p2
p143
q3 q412
p214
p3
and
δ0a0 δ
1
2d+1
δ−
1
2d−1 δc0
q1q234
p432
p1
q123 q4
p4
p321
III.3. SKEIN RELATIONS 83
Obviously, these peculiar modules are loop-type. Also, reversing all arrows, swapping pis
and qis and reversing the Alexander grading leaves both of them invariant. Doing this to
the mapping cone gives us maps between the mirrors of the two tangles, but in the opposite
direction. 
Remark 3.5. It is interesting to compare the “figure-8” curve to the local Heegaard
diagram for a singular crossing b in [OSS07], as the number of generators agree for the
second loop above, up to an additional tensor factor. Also note that the proposition above
gives rise to an exact triangle similar to the one in [OS07]. Moreover, we can write the
n-twist tangle Tn from figure 41a, with both strands oriented upwards, as a complex in the
objects b and . Indeed, cancelling the identity components in the following complex
gives us a loop representing Tn.
b2−n b4−n · · · bn dn
a1−n a3−n · · · an−1
c1−n c3−n · · · cn−1
b−n b2−n · · · bn−2 bn
q412
q3
q412
q3
q412q3
p2
p143
p14
p2
p143
p14
p2
p143
p1
p214
p3 q41
p214
p3 q41
p214
p3
q4
q2 q341
q2 q341
1
q2 q341
1 1
p43+q12
p21+q34
Similarly, we can obtain a complex for T−n by applying the mirror operation. Furthermore,
it is also easy to find such complexes for other orientations of T−n and Tn. Note that these
complexes look very much like the ones we get in Bar-Natan’s Khovanov homology of tangles
[Bar04]. We assume that every tangle can be written as a complex in the two objects b and
, or the two objects and , depending on the orientation. In fact, we can iteratively
use the type AA glueing structure from theorem 2.5 together with the skein exact sequence
from theorem 3.1 to locally modify tangles until we obtain a complex of peculiar modules of
trivial and 1-crossing tangles, up to a large number of tensor factors from glueing. Then, one
(only) needs to get rid of these extra factors. For example, in the case of the (2,−3)-pretzel
tangle, this is indeed possible; we may write it as a complex of elementary tangles. This is
actually how I originally found the curves in figure 34 in the Fukaya category setting (see
section III.5) before recovering them from tangle Heegaard diagrams.
It is also interesting to compare our “figure-8” curve to the curve that Hedden, Herald, Kirk
associate with a trivial tangle in [HHK13, figure 10] in the context of instanton knot Floer
homology in the pillowcase, which is the following:
84 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
III.4. Symmetry relations for CFT∂
We start with a result which can be viewed as a categorification of the 4-term relations
between the Alexander polynomials for different sites from corollary I.3.2.
Proposition 4.1. Let T be a 4-ended tangle and consider CFT∂(T ). After passing to
the quotient algebra A∂ /(q1 = q2 = q3 = q4 = p1 = 0) and taking homology, we obtain the
following chain complex
ĤFT(T, d) ĤFT(T, a)
ĤFT(T, c) ĤFT(T, b)
Φp4
Φp43
Φp432
Φp3
Φp32 Φp2
where we regard the maps Φ· as homomorphisms between the four vector spaces ĤFT(T, s),
s ∈ {a, b, c, d}. This chain complex is null-homotopic. By symmetry, the corresponding
statements also hold when we cyclically permute sites and algebra elements or swap the roles
of the pis and qis.
Proof. The above complex can be identified with the type D module of the following
bordered sutured structure on the tangle complement.
a
b
c
d
T
Taking homology of this complex is the same as pairing this type D module with the type
A-module computed from the following diagram.
a b c d
However, if we glue these two bordered sutured manifolds together, we obtain a sutured
manifold which is not taut, so its sutured Floer homology vanishes by [Juh06a, proposi-
tion 9.18]. 
The maps Φpi and Φqi are, by definition, invariants of the tangle T . We do not consider
any naturality issues here, so this essentially just means that the ranks of all bigraded parts
of the maps are invariants. For loop type CFT∂(T ), those ranks are simply the number of
arrows labelled by the corresponding pis or qis.
III.4. SYMMETRY RELATIONS FOR CFT∂ 85
Proposition 4.2. With the notation as in the previous proposition, let
rkδ(p1) := rk
(
Φp1 : ĤFTδ(T, a)→ ĤFTδ+ 1
2
(T, d)
)
and similarly for the other pi and all qi. Then
rkδ(p1) = rkδ(q2) = rkδ(p3) = rkδ(q4) and rkδ(q1) = rkδ(p2) = rkδ(q3) = rkδ(p4).
This result, together with the symmetry relations for generators, tempts us to conjecture
the following.
Conjecture 4.3 (δ-graded mutation invariance). Let T be a 4-ended tangle and T ′
obtained from T by switching two or all four opposite sites of T . Then,
CFT∂(T ) ∼= CFT∂(T ′)
as δ-graded invariants.
Together with the glueing theorem, this would imply δ-graded mutation invariance of
link Floer homology (conjecture 0.3).
Proof of proposition 4.2. The proof is very similar to the previous one. By sym-
metry, we only need to prove the first set of equalities. Let us consider the identity
rkδ(p1) = rkδ(p3) first. The maps Φp1 and Φp3 can be computed from the bordered su-
tured structure on the tangle complement shown on the left; we either use the dotted α-arcs
or the solid ones:
a
b
c
d
T −→ T
Computing the homology of Φp1 , resp. Φp3 , i. e. computing sum of the dimensions of the
kernel and cokernel in each (bi)grading, corresponds to pairing these two bordered sutured
manifolds with the one below. In both cases, the resulting sutured manifold is the same, so
the (bi)graded sutured Floer homologies agree.
b resp. d c resp. a
We now use proposition II.1.8 to see that the dimensions of the δ-graded parts of the domain
and codomain agree, so the ranks are also the same.
Similarly, we can show rkδ(q2) = rkδ(q4). Note that the two sutured manifolds for this iden-
tity and the previous one are very similar: In one case, we can push the meridional sutures
through the tangle to see that we have the same set of sutures up to reversal of orientation.
86 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Then, the δ-graded sutured Floer homologies agree by [FJR09, proposition 2.14]. In the
other case, the two manifolds contain one pair of meridional sutures on the open tangle
components and are obtained from one another by deleting one and adding the other. In
this case, we can consider exactly the same decomposing surface as in the proof of proposi-
tion II.1.8 and use [Juh06b, proposition 8.6] to see that, again, the δ-graded sutured Floer
homologies are the same. Now we can argue as before, by using proposition II.1.8. 
Remark 4.4. The symmetries in the previous proposition do not hold for bigraded
ranks. (As a counterexample, consider the computation for the (2,−3)-pretzel tangle in
example 1.18.) The homologies of the maps, considered as differentials, agree as bigraded
vector spaces, but in order to identify the tangle Floer homologies of opposite sites, we need
to reverse the Alexander grading.
Remark 4.5. In view of the calculations for the (2,−3)-pretzel tangle (theorem II.3.14
and example 1.18) and the relation between the non-glueable tangle Floer homology for
opposite sites II.1.8, one might be tempted to conjecture the following, bigraded version of
conjecture 4.3:
Let T be a 4-ended tangle and T ′ obtained from T by switching opposite
sites of T and reversing the orientation of all tangle strands. Then,
CFT∂(T ) ∼= CFT∂(T ′)
as bigraded invariants, where we identify the Alexander gradings of the two
open tangle strands.
However, it turns out, this version is too strong. For this, consider a 4-ended tangle T
oriented such that the outward pointing ends are next to one another. Then the conjecture
above, together with a glueing theorem, would imply that mutation in the plane (Mutz)
about any such tangle T leaves bigraded knot Floer homology invariant. This is because for
this particular orientation and mutation axis, the orientation of the open tangle strands of
the mutating tangle needs to be reversed before we can glue it back in. However, we can
write mutation about one axis as mutation about another, by introducing twists:
Mutz

b
b
Γ
 = bb
Γ
=
b
b
Γ
So we deduce that mutation about the vertical axis also leaves bigraded knot Floer homology
invariant if the orientation of the mutating tangle is such that the outward pointing ends are
opposite each another. This, however, cannot be true, as the Kinoshita-Terasaka/Conway
pair (example II.3.15) shows, so the conjecture above is indeed too strong.
Question 4.6. Let T be a 4-ended tangle which is symmetric about one mutation axis.
Can we always find an orientation such that CFT∂ is bigraded mutation invariant with
respect to the other two axes?
III.5. CFT∂ AND THE WRAPPED FUKAYA CATEGORY OF THE 4-PUNCTURED SPHERE 87
III.5. CFT∂ and the wrapped Fukaya category of the 4-punctured sphere
In this last section, we explore the relationship between the peculiar invariants of 4-ended
tangles and the wrapped Fukaya category Fuk of the 4-punctured sphere. Much of this is
still work in progress, so there are still plenty of open questions, some of which we discuss
in an outlook at the end.
What is Fuk? The category Fuk is an A∞-category which, roughly speaking, encodes
the Lagrangian intersection theory of (exact) Lagrangians on a 4-punctured sphere (equipped
with a certain 1-form). We will not discuss the technical details of its construction here,
but refer the interested reader to [AAEKO, section 4] instead. The objects in Fuk are
the Lagrangians and the morphisms essentially correspond to intersection points of these.
The A∞-composition maps are defined using certain holomorphic polygons connecting those
intersection points. As usual in Lagrangian intersection theory, we need to perturb the
Lagrangians by some Hamiltonian flow when computing morphisms. Here, we restrict our-
selves to certain Hamiltonians which are “quadratic at infinity”, which effectively means that
a Lagrangian approaching a puncture is wrapped infinitely many times around the puncture
by such a Hamiltonian perturbation – hence the name wrapped Fukaya category. This is
illustrated in figure 44.
In [AAEKO], Abouzaid et al. study the wrapped Fukaya category Fuk(S2, n) of the n-
punctured sphere for n ≥ 3. They introduce a finite auxiliary A∞-category A which gener-
ates Fuk(S2, n) and describe some structure maps of A, namely all higher compositions up to
length n. Then, using a Hochschild homology argument, they show that these compositions
uniquely characterise A, and hence Fuk(S2, n), up to homotopy, see [AAEKO, theorem 4.1].
Relationship with peculiar modules. For the case n = 4, we will show that
Fuk := Fuk(S2, 4) and the category of peculiar modules pqMod are closely related. In the-
orem 5.10, we construct A∞-functors M and L between pqMod and the category TwFuk,
the triangulated enlargement of Fuk, also known as the category of twisted complexes. For
more background on the category TwFuk, twisted complexes and A∞-categories in general,
see [Sei08, section I.1a–d and I.3l], [AAEKO, remark 4.2] and example A.18.
Given a 4-ended tangle T , L(CFT∂(T )) is an object in TwFuk which is an invariant of T
up to homotopy. Given also a site s of T , we can interpret ĤFT(T, s) as the Lagrangian
intersection homology of L(CFT∂(T )) and a certain generator Ls of TwFuk associated with
s. Moreover, computations suggest that the two functorsM and L actually set up an equiv-
alence of categories.
For the construction ofM and L, it is essential that we understand the composition maps in
TwFuk. This is equivalent to understanding the finite category A, since by the generation
result from [AAEKO], TwFuk agrees with TwA.
The auxiliary category A. Following [AAEKO, section 2], we build the A∞-
category A from an ordinary category A, which we define now.
88 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Definition 5.1. Let A be a category with n objects L1, . . . , Ln. The morphism spaces
between these objects are as follows:
Mor(Li, Lj) :=

k[xi, yi]/xiyi for i = j
k[xi+1]ui,i+1 = ui,i+1k[yi] for j = i+ 1
k[yi−1]vi,i−1 = vi,i−1k[xi] for j = i− 1
0 otherwise
,
where indices are taken modulo n. Each morphism space comes with a natural basis, namely
{1, xli, yli|l ≥ 1}, {xli+1ui,i+1 = ui,i+1yli|l ≥ 0} and {yli−1vi,i−1 = vi,i−1xli|l ≥ 0},
respectively. In the following, when we talk about morphisms, we mean those basis elements.
By sequences of morphisms of length k, we mean elements of the induced basis of the tensor
space
Mor(Lik−1 , Lik)⊗ · · · ⊗Mor(Li0 , Li1).
For k = 2, this is the domain of the composition map µ2 for A, which we describe later;
for arbitrary k, this will be the domain for higher compositions in A. We sometimes drop
the index when it is clear from the context. We follow the convention in [Sei08] to read
sequences of morphisms from right to left.
Graphical representation of morphisms in A and A. For explicit calculations, it
is helpful to depict morphisms and sequences thereof graphically. The basic objects Li are
represented by infinitely many levels (i. e. horizontal lines, but we never actually draw them),
numbered cyclically from 1 to n, each of them at a constant positive distance from its two
neighbours – like on a stave. A morphism from Li to Lj is represented by a straight line
from a level corresponding to Li to the closest one that corresponds to Lj . So by definition,
such a line either moves up by one, stays constant or moves down a level. Next, we label
the line by the morphism, but the advantage now is that we can drop the indices, without
loosing information. Furthermore, if the level changes, we only need to record the exponent
l in xlu or vxl. For simplicity, we often drop the label completely if this exponent is 0. For
sequences of morphisms, we just concatenate the lines representing the morphisms in the
sequence in the specified order. In this graphical representation, we read sequences from left
to right.
We indicate the numbering of the levels by specifying the level of the starting point of the
first morphism and use the convention that the numbering increases from bottom to top.
Often, we even drop this information, since the multiplication maps are symmetric in the Li.
Example 5.2. The following table shows some morphisms and their graphical notation
as described above. All morphisms start at the index 1.
u1,2 x2u1,2 y
2
1 11 x
4
1 v1,n y
2
1v1,n
1 y2 1 x4 2
III.5. CFT∂ AND THE WRAPPED FUKAYA CATEGORY OF THE 4-PUNCTURED SPHERE 89
The sequence s = (x3u2,3, u1,2, x1, v2,1, v3,2, y3) is represented by
3 y
x
1
In the case n = 4, we will see below that µ of this sequence is the morphism x3.
Definition 5.3 (composition in A). Most compositions are already implicit in the def-
inition of the morphism spaces as rings and bimodules. The only remaining composition
are
µ
 · ·
 = 0, µ
 · ·
 = 0,
µ
(
k l
)
= yk+l+1 and µ
(
k l
)
= xk+l+1 for k, l ≥ 0.
For completeness, we also describe the other obvious compositions graphically:
µ
(
k
xl
)
= k + l and µ
(
xl
k
)
= k + l for k, l ≥ 0,
µ
(
yl
k
)
= k + l and µ
(
k
yl
)
= k + l for k, l ≥ 0.
Definition 5.4 (higher compositions in A). The A∞-category A is built from A by
adding higher composition maps. They are recursively defined by
µ
. . . , µ2(b, ) , , . . . ,︸ ︷︷ ︸
n−2
, µ2
(
, a
)
, . . .
 := µ(. . . , b, a, . . . ) and
µ
. . . , µ2(b, ) , , . . . ,︸ ︷︷ ︸
n−2
, µ2
(
, a
)
, . . .
 := µ(. . . , b, a, . . . ),
where a and b are some morphisms, all sequences are composable and the two µ2-terms on
the left hand sides are all non-zero. In other words, all non-zero A∞-terms are obtained by
“expanding” non-zero 2-term sequences by n-term sequences of us and n-term sequences of
vs. In particular, the differential on A vanishes.
Proposition 5.5. The maps µ above define a unital A∞-structure on A.
Proof. To my knowledge, this was first proven in [Boc07, proposition A.11]. For a
more concise and elegant proof, which simply exhibits cancelling pairs in the A∞-relations,
see [HKK14, proposition 3.1]. 
Gradings on A. In [AAEKO], the morphisms in A carry a Z-grading deg defined by
deg(ui,i+1) = deg
(
i
)
= pi+1 and deg(vi,i−1) = deg
(
i
)
= qi,
90 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
where p1, . . . , pn and q1, . . . , qn are some odd integers, satisfying the relations
(11) p1 + · · ·+ pn = (n− 2) = q1 + · · ·+ qn.
It is easy to see that this extends to a well-defined grading on all basic morphisms using
deg(µ(b, a)) = deg(a) + deg(b).
The relations (11) imply that µk decreases the grading by (k− 2). As usual, we extend this
grading to TwA by setting
(12) deg(f : Li[x]→ Lj [y]) = deg(f) + y − x.
for homogeneous f .
However, here, we replace deg by several, slightly different gradings which correspond to the
gradings on our Heegaard Floer invariants. For this, let us treat p1, p2, p3, p4, q1, q2, q3 and
q4 as formal variables in some free Abelian monoid.
b b
bb
L1
L2
L3
L4
1
2 3
4
p1
q1
p2
q2
p3
q3
p4
q4
u4,1
v1,4
x1
y4
u1,2
v2,1
x2
y1
u2,3
v3,2
x3
y2
u3,4
v4,3
x4
y3
Figure 43. The A∞-category of the 4-punctured sphere in a nutshell. Compare
this to figure 31.
Definition 5.6. Fix an orientation of the four punctures, as if they were endpoints of
a 4-ended tangle; in other words, two punctures are labelled “in” and the other two “out”,
compare with definition 1.7. Define the Alexander grading by setting A = deg with
pi = qi = +1 if the ith puncture is labelled “out” and pi = qi = −1 otherwise. As for CFT∂ ,
we sometimes denote the Alexander grading of a basic object by LA(∗)j . Similarly, we can
define a multivariate Alexander grading by separating the four punctures into two pairs of
oppositely oriented punctures. Note that
(13) p1 + p2 + p3 + p4 = 0 = q1 + q2 + q3 + q4,
so the structure maps µk preserve the Alexander grading.
III.5. CFT∂ AND THE WRAPPED FUKAYA CATEGORY OF THE 4-PUNCTURED SPHERE 91
Definition 5.7. We define a 12Z-grading on morphisms in A, which we call δ-grading,
by setting δ = deg with pi = qi = 12 . In this case,
(14) p1 + p2 + p3 + p4 = 2 = q1 + q2 + q3 + q4,
so the structure maps µk decrease grading by (k−2). Given an orientation on the punctures,
the homological grading h is then defined by h = 12A − δ. This is equivalent to setting
pi = qi = 0 if the ith strand is labelled “out” and −1 otherwise.
Remark 5.8. We can interpret morphisms and sequences thereof in terms of paths in
the 4-punctured sphere, see figure 43. In fact, this is where the proofs of proposition 5.5
come from.
Definition 5.9. Consider the opposite algebra A∂op of A∂ , i. e. the algebra defined by
A∂op := {(a)−1|a ∈ A∂}, (a)−1.(b)−1 := (b.a)−1, ι1.(a)−1.ι2 := (ι2.a.ι1)−1,
where ι1, ι2 ∈ I∂ . If deg is a grading on A∂ , it induces a grading on A∂op by
deg((a)−1) = −deg(a)
for homogeneous a ∈ A∂ . We also define two I∂-algebra homomorphisms
A∂op×A∂ → A∂op and A∂ ×A∂op → A∂op .
If a, b ∈ A∂ are paths in the quiver algebra, define
((a)−1, b) 7→ (a)−1.b :=
(c)−1 if ∃path c : bc = a0 otherwise b ca
Note that if c exists, it is unique. Similarly, let
(b, (a)−1) 7→ b.(a)−1 :=
(c)−1 if ∃path c : cb = a0 otherwise c ba
Then extend both maps linearly. Obviously, they define I∂-algebra homomorphisms.
Theorem 5.10. Let TwFuk be the triangulated enlargement of the Fukaya category of the
4-punctured sphere. As discussed in the introduction of this section, we can identify TwFuk
with the A∞-category TwA of twisted complexes of A, see example A.18. Let pqMod be
the dg category of peculiar modules, considered as an A∞-category. We define a covariant
functor
M : TwFuk→ pqMod
as follows. Given a twisted complex L ∈ ob TwFuk,M(L) is defined as an I∂-module by
M(L) :=
⊕
i=1,2,3,4
ιi.Mor(Li, L).
The structure map ∂M(L) :M(L)→ A∂ ⊗M(L), given by
f 7→ 1⊗ µtw1 (f) +
∑
n≥1
pn ⊗ µtwn+1(f, u, . . . , u︸ ︷︷ ︸
n
) + qn ⊗ µtwn+1(f, v, . . . , v︸ ︷︷ ︸
n
),
turns (M(L), ∂M(L)) into a well-defined peculiar module.
92 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
Moreover, given a sequence of morphisms of twisted complexes (ar, . . . , a1) : L(0)→· · ·→L(r),
we define a map
Mr(ar, . . . , a1) :M(L(0))→ A∂ ⊗M(L(r)),
f 7→1⊗ µtwr+1(ar, . . . , a1, f) +
∑
n≥1
pn ⊗ µtwn+r+1(ar, . . . , a1, f, u, . . . , u︸ ︷︷ ︸
n
)
+
∑
n≥1
qn ⊗ µtwn+r+1(ar, . . . , a1, f, v, . . . , v︸ ︷︷ ︸
n
).
We also have a covariant functor
L : pqMod→ TwFuk,
defined as follows. Given a peculiar module (M,∂M ) ∈ ob pqMod, we define a twisted
complex
L(M) :=
⊕
j=1,2,3,4
(
ιj .A∂op⊗I∂M
)
⊗ Lj
with structure map ∂L(M) : L(M)→ L(M) given by
b⊗m⊗ Lj 7→ b.∂M (m)⊗ idLj +p.b⊗m⊗ uj,j+1 + q.b⊗m⊗ vj,j−1.
Furthermore, if a : M → A∂ ⊗M ′ is a morphism of peculiar modules, we define a morphism
L(a) : L(M)→ L(M ′) by
b⊗m⊗ Lj 7→ b.a(m)⊗ idLj .
We set L(ar, . . . , a1) = 0 for any sequence of morphisms (a1, . . . , ar) with r > 1.
Remark 5.11. The two functors are compatible with the various gradings on both sides.
As usual, the gradings on a tensor product are given by the sum of the gradings of all factors;
the gradings on morphism spaces is as defined in equation (12).
Example 5.12. The image of L1 underM is given by
· · · d a d a b a b · · ·
q1
q234
p1
p432
q1
q234
p1
p432
q341
q2
p143
p2
q341
q2
p143
p2
which corresponds to the curve in figure 44. Similarly, if we (formally) apply L to (M,∂) =
(〈a〉, 0), we obtain
· · · L1 L2 L3 L4 L1 L2 L3 L4 L1 · · ·u u u u u v v v v v
µtw1 of this complex is non-zero, but note that (〈a〉, 0) is not a peculiar module either.
Proof of theorem 5.10. The infinite sums in the definition ofM are actually finite,
since by definition of the A∞-structure all terms for n > 3 vanish. (However, we stick to the
notation above as it is easier.) Let us check that ∂M(L) satisfies the required ∂2-relation.
The coefficient of 1 vanishes, since µtw1 is a differential. The coefficient of pn is given by∑
i+j=n
i,j≥0
µtwj+1(µ
tw
i+1(f, u, . . . , u︸ ︷︷ ︸
i
), u, . . . , u︸ ︷︷ ︸
j
)
=
∑
i+j+k+1=n
i,j,k≥0
µtwi+k+2(f, u, . . . , u︸ ︷︷ ︸
i
, µtwj+1(u, . . . , u︸ ︷︷ ︸
j+1
), u, . . . , u︸ ︷︷ ︸
k
).
III.5. CFT∂ AND THE WRAPPED FUKAYA CATEGORY OF THE 4-PUNCTURED SPHERE 93
b b
bb
M(L1)
L1
L2
L3
L4
Figure 44. The curve representing the peculiar moduleM(L1)
The term µtwj+1(u, . . . , u) = µj+1(u, . . . , u) vanishes for all j except j = 3, in which case it is
the identity. Since µ and hence also µtw is unital, this implies that the term above vanishes
for all n except n = 4, in which case there is just one summand, namely
µtw2 (f, µ
tw
4 (u, u, u, u)) = µ
tw
2 (f, 1) = µ2(f, 1) = f.
We can argue similarly for the coefficient of qn. SoM is well-defined on objects.
Next we need to check thatM satisfies the A∞-relations
D(M(ar, . . . , a1)) +
∑
0<l<r
M(ar, . . . , al+1) ◦M(al, . . . , a1)
=
∑
1≤i<j≤r
M(ar, . . . , µtwj−i+1(aj , . . . , ai), . . . , a1)
for all sequences of morphisms (a1, . . . , ar) and r ≥ 1. This follows in the same way as above,
except that there is no contributing term as
µtwr+2(ar, . . . , a1, g, µ
tw
4 (u, u, u, u)) = µ
tw
r+2(ar, . . . , a1, g, 1) = 0.
Next, consider L. The only non-zero terms in µtw1 (∂L(M)) correspond to µ2(id, id),
µ4(u, u, u, u), µ4(v, v, v, v), µ2(id, u), µ2(u, id), µ2(id, v), µ2(v, id). The first three cases
constitute the identity component idLi , which vanishes as ∂2M = p
4 + q4. The terms in the
fourth and fifth case cancel each other, and so do those in the last two. The δ-grading defines
a filtration on L(M) which ensures that the structure maps are upper-triangular.
Since pqMod has no higher multiplications and L vanishes on morphism sequences of length
greater than 1, there are only two non-trivial A∞-relations to check:
µtw1 (L(a)) = L(D(a)) and µtw2 (L(a1),L(a2)) = L(a1 ◦ a2),
both of which are straightforward to check. 
Example 5.13. Figure 45 shows the images of the peculiar modules corresponding to
some basic tangles under the functor L after doing some cancellation. Conversely, we can
recover the original peculiar modules by applyingM to these twisted complexes and, again,
doing some cancellation. This seems to be sufficient computational evidence to justify the
following conjecture.
Conjecture 5.14. The functors M and L give rise to an equivalence of A∞-categories.
94 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
b
δ0b0 δ0d0
p43+q12
p21+q34
δ0a0 δ
1
2d+1
δ
1
2 b−1 δ0c0
p432
q341
p1
q4
p3
q2
p214
q123
δ0a0 δ−
1
2d−1
δ−
1
2 b+1 δ0c0
q1
p2
q234
p321
q412
p143
q3
p4
δ0a0 δ−
1
2 b+1
δ
1
2 b−1 δ0c0
q341q2
p2
p143
q3 q412
p214
p3
δ−1L02 δ
− 1
2L+11
δ−
1
2L+13 δ
0L04
δ−
1
2L−12 δ
0L01
δ0L03 δ
− 1
2L+14
δ−
1
2L+12 δ
−1L01
δ−1L03 δ
− 1
2L−14
δ−
1
2L−12
δ−
1
2L+12
x+y
Figure 45. Some basic rational tangles, their peculiar modules and the images
thereof under L. The superscripts of the generators specify the Alexander grading.
The last column shows one of the two possible invariants for the singular crossing
from proposition 3.4.
As a first step towards a proof of this conjecture, we offer the following proposition. Its
proof is an exercise in cancellation. The conjecture should follow from studying functoriality
properties of cancellation and a reformulation of these in the A∞-context.
Proposition 5.15. Let T be a 4-ended tangle. Then the tangle Floer complex ĈFT(T, a)
is bigraded chain homotopic to(
Mor
(
L1,L
(
CFT∂(T )
))
, µtw1
)
.
By symmetry, the corresponding statement holds for all other three sites.
Proof. Let M := CFT∂(T ). On the one hand, ĈFT(T, a) is obtained from ι1.M by
setting pi = qi = 0 for i = 1, 2, 3, 4. On the other hand, we can view ι1.M as a subspace of
(15) Mor(L1,L(M)) =
⊕
j=1,2,3,4
ιj .A∂op⊗I∂M ⊗Mor(L1, Lj)
via
ι1.m 7→ (1)−1 ⊗m⊗ 1L1 .
The differential on this complex is given by
µtw1 : a⊗m⊗ g 7→ a.∂M (m)⊗ g + p.a⊗m⊗ µ2(u, g) + p3.a⊗m⊗ µ4(u, u, u, g)
+ q.a⊗m⊗ µ2(v, g) + q3.a⊗m⊗ µ4(v, v, v, g)
The idea is to cancel all other generators in this chain complex using the cancellation
lemma A.19. For this, we split the vector space from (15) into the following three com-
ponents:
Z =
〈
(1)−1 ⊗m⊗ 1L1
∣∣m ∈ ι1.M〉
III.5. CFT∂ AND THE WRAPPED FUKAYA CATEGORY OF THE 4-PUNCTURED SPHERE 95
Y1 =
〈
ι1.p
−i ⊗M ⊗ yj
∣∣∣∣∣ i > 0, j ≥ 0
〉
⊕
〈
ι4.p
−i ⊗M ⊗ j
∣∣∣∣∣ i > 0, j ≥ 0
〉
⊕
〈
ι1.q
−i ⊗M ⊗ xj
∣∣∣∣∣ i > 0, j ≥ 0
〉
⊕
〈
ι2.q
−i ⊗M ⊗ j
∣∣∣∣∣ i > 0, j ≥ 0
〉
Y2 =
〈
ι2.p
−i ⊗M ⊗ j
∣∣∣∣∣ i ≥ 0, j ≥ 0
〉
⊕
〈
ι1.p
−i ⊗M ⊗ xj
∣∣∣∣∣ i ≥ 0, j > 0
〉
⊕
〈
ι4.q
−i ⊗M ⊗ j
∣∣∣∣∣ i ≥ 0, j ≥ 0
〉
⊕
〈
ι1.q
−i ⊗M ⊗ yj
∣∣∣∣∣ i ≥ 0, j > 0
〉
µtw1 (Z) ⊆ Z, so in particular the map c : Z → Y2 vanishes. In fact, the restriction map µtw1 |Z
only counts the identity components of ∂M , so it agrees with the differential on ĈFT(T, a).
So if we can show that f = µtw1 |Y1→Y2 is an isomorphism, we are done.
Let us write f = f0 +f1, where f1 is the first component of µtw1 . We claim that f0 sets up an
identification of Y1 with Y2 as vector spaces. Indeed: the second component of µtw1 sets up a
bijection on the level of generators between the first two summands of Y1 and Y2; similarly,
the fourth component of µtw1 identifies the other two summands; finally, the third and fifth
components of µtw1 vanish on Y1.
Let g0 be the inverse of f0 and define
g := g0 + g0f1g0 + g0f1g0f1g0 + g0f1g0f1g0f1g0 + . . .
g is a well-defined homomorphism, since g0 lowers the δ-grading of the last tensor factor, f1
preserves it and the δ-grading on Mor(L1, Lj) is bounded below. Obviously, g is an inverse
of f . 
Outlook. Ideally, we would like to reformulate the pairing theorem 2.5 for our peculiar
invariants in terms of Lagrangian intersection homology as follows.
Conjecture 5.16. Let L, T1 and T2 be as in theorem 2.5. Then ĈFL(L) is bigraded
chain homotopic to (
Mor
(
L
(
CFT∂(m(T1))
)
,L
(
CFT∂(T2)
))
, µtw1
)
.
It would be interesting to see if one can define a functor similar to L for type AA
structures and what the image of the glueing structure P from theorem 2.5 would look like
under this functor.
Question 5.17. Does L send loop-type peculiar modules to twisted complexes represent-
ing closed curves and vice versa?
Remark 5.18. According to a classification result from [HKK14, theorem 4.3], any
twisted complex in a (partially) wrapped Fukaya category of a punctured surface either
represents a closed curve on this surface or an arc connecting two punctures (up to the
issue of local systems). If such a classification is also true for our fully wrapped Fukaya
category Fuk we might be able to show that all peculiar modules of tangles are loop-type
(question 1.21), using the following argument: If a peculiar moduleM corresponds to an arc,
96 III. PECULIAR INVARIANTS FOR 4-ENDED TANGLES
the intersection homology with some Li is an infinite-dimensional vector-space. If at the same
time M were homotopic to CFT∂(T ) for some 4-ended tangle, then, by proposition 5.15,
this vector space would be isomorphic to ĤFT(T, si) for the site si corresponding to Li. But
the tangle Floer homology is always finite-dimensional, so we have a contradiction.
APPENDIX A
Algebraic structures from dg categories
In chapters II and III, we often work in categories of various algebraic structures, namely
type A, type AA, type D and curved type D structures. In all four settings, we often want to
simplify these structures by replacing them by homotopy equivalent ones. The main goal of
this appendix is to develop some tools for dealing with this problem, namely the cancellation
lemma (A.19) and the clean-up lemma (A.21). The former can be used to reduce the number
of generators of an algebraic structure, the latter for making the structure maps “look nicer”,
essentially by changing the basis.
In the category of ordinary chain complexes, both tools will be familiar to the reader as
easy exercises in linear algebra. So it might not be too surprising that they also work in
quite general settings. We will spend the first part of this appendix explaining a general
construction which turns any differential graded category into another such category in which
the lemmas hold in some generality sufficient for our purposes, see definitions A.4 and A.5.
Next, we show that the various different algebraic structures mentioned above arise naturally
from this general construction. Finally, we state and prove the cancellation and clean-up
lemmas in this general framework.
For simplicity, we only work over the field F2 = Z/2, so we do not need to keep track of
signs. However, with the correct sign conventions, all statements should also hold over fields
of arbitrary characteristic.
Definition A.1. Let Com be the category of Z-graded chain complexes over F2 and
grading preserving chain maps between them. A differential graded (dg) category C
over F2 is an enriched category over Com. To spell this out more explicitly, the hom-objects
are Z-graded F2-vector spaces,
Mor(A,B) =
⊕
i∈Z
Mori(A,B)
endowed with differentials
∂i : Mori(A,B)→ Mori−1(A,B),
i. e. vector space homomorphisms satisfying ∂i−1∂i = 0 and
(16) ∂ ◦m = m ◦ (∂ ⊗ id + id⊗∂),
where
m : Mori(A,B)⊗Morj(B,C)→ Mori+j(A,C)
denotes composition in C, which is associative and unital. For more details on enriched
categories, see for example [Rie14]. Note that the identity morphisms have degree zero and
lie in the kernel of ∂.
97
98 A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES
Definition A.2. [Rie14, definition 3.4.5]. Given an enriched category C over some
monoidal category V, the underlying ordinary category C0 of C has the same objects as
C and its hom-sets are defined by
C0(A,B) := MorV(1V , C(A,B)).
Example A.3. Let C be a dg category. The unit in Com is the complex 0 → F2 → 0,
supported in homological degree 0, and the morphisms in Com are grading preserving. Hence,
the hom-sets of C0 are those elements in the kernel of ∂0.
Consider the enriched category H∗(C) over the category of graded vector spaces and grading
preserving morphisms between them, obtained from C by replacing the hom-objects by their
homologies with respect to the differential ∂. By passing to the underlying ordinary category,
we pick out the degree 0 morphisms in H∗(C). Therefore, we denote this category by H0(C).
Since the hom-sets in H0(C) are just quotients of those in C0, we now get the usual notions
of chain homotopies between morphisms and objects. The reason why we need to pass to
the underlying category is that otherwise, two objects could be (chain) isomorphic through
grading shifting morphisms.
Definition A.4. (cp. [Bar04, section 6]) Given a dg category C, we define another dg
category Mat(C) as follows. Its objects are formal direct sums⊕
i∈I
Oi[ni],
where I is some index set and Oi[ni] denotes the object Oi ∈ obj(C) with a formal grading
shift by an integer ni. Morphisms are given by
Morn(
⊕
i∈I
Oi[ni],
⊕
j∈J
Oj [nj ]) :=
⊕
(i,j)∈I×J
Morn+ni−nj (Oi, Oj).
Compositions and differentials in Mat(C) are induced by those in C.
Definition A.5. Given a differential graded category C, we define an auxiliary category
Cxpre(C), the category of pre-complexes, which is an enriched category over the category
of Z-graded vector spaces and grading preserving morphisms between them. Its objects are
pairs (O, dO), where O ∈ obj(C) and dO ∈ Mor−1(O,O). The hom-objects are the same as
in C,
Mor((O, dO), (O
′, dO′)) = Mor(O,O′),
viewed as Z-graded vector spaces. On these, we can define a map
D : Mori((O, dO), (O
′, dO′))→ Mori−1((O, dO), (O′, dO′))
by setting
D(f) := dO′ ◦ f + f ◦ dO + ∂(f).
We would like D to be a differential in order to turn Cxpre(C) into a dg category. However,
this only works in general if we restrict ourselves to a full subcategory of Cxpre(C):
It is easy to check that D is always compatible with multiplication in the sense of (16). So
D is a differential iff
D2(f) = (d2O′ + ∂(dO′)) ◦ f + f ◦ (d2O + ∂(dO))
A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES 99
vanishes. This is, of course, the case for the full subcategory Cx0(C) of Cxpre(C) consisting of
those objects (O, dO) for which
(∗) d2O + ∂(dO)
vanishes. However, in some situations, other conditions on (∗) also work. For example, if we
replace Com by the category of Z/2-graded chain complexes, we can restrict to those objects
(O, dO) for which (∗) is equal to the identity. Also, if the hom-objects are bimodules over
an algebra A, we can ask (∗) to be equal to a. idO for a fixed central algebra element a (of
degree −2) which commutes with all morphisms f . In both cases, D will be a differential.
In any of these cases, the resulting category is a differential graded category again and we call
any such full subcategory a category of complexes, denoted by Cx∗(C), where ∗ ∈ {0, 1, a}
is the value of (∗).
Remark A.6. As usual, we can associate a directed graph to a category, where objects
correspond to vertices and arrows to morphisms. In the same way, we can think of complexes
in Cx∗(Mat(C)) as graphs. We often label the arrows by the morphisms.
The point of the construction above is that after choosing a basis, we can interpret
the categories of type D, type A, type AA and curved type D structures as instances of
Cx∗(Mat(C)) for suitable choices of relatively simple differential graded categories C. But let
us start with an even simpler example: ordinary chain complexes.
Note of warning. In the following examples, our definitions only coincide with the
usual ones after passing to the underlying ordinary categories, see example A.3. The advan-
tage of our point of view is that the conditions we usually impose on morphism and chain
homotopies for various algebraic structures arise naturally by viewing those morphisms as
elements of chain complexes.
Example A.7 (ordinary chain complexes over F2). Let C be the category with a single
object • in grading 0, Mor(•, •) = F2 and vanishing differential. Then (the underlying
ordinary category of) Cx0(Mat(C)) is Com.
Example A.8 (type D modules over dg F2-algebras). Let A be a differential graded
algebra over F2. Let C be the category with a single object • and morphisms being elements
in A. Composition is multiplication in A and the differential ∂ is induced by the differential
on A. We define the category of type D modules by Cx0(Mat(C)). (Again, note that we
need to pass to the underlying ordinary category to obtain the definitions in [Zar09] and
[LOT08].)
Example A.9 (type D modules over dg I-algebras). Let us assume that A is an algebra
over some ring I ⊆ A of idempotents and fix a basis {ij}j∈J of idempotents of I, where J is
some index set. Let C be the category with one object for each basis element of I, and for
any two such elements i1 and i2, let Mor(i1, i2) := i1.A.i2, viewed as a quotient of A. Again,
composition is multiplication in A and the differential ∂ is induced by the differential on A.
We define the category of type D modules over dg I-algebras by Cx0(Mat(C)).
100 A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES
Remark A.10. A priori, the definition in the previous example depends on a choice of
basis for I. In the examples that we see in chapters II and III, there is a natural choice of
such a basis, so this is not an issue.
However, we can replace C above by the enlarged category CI , where there is an object for
every element in I. Then C is a full subcategory of CI and it is not hard to see that Mat(C)
andMat(CI) are equivalent. Now, the construction of the category of complexes is functorial
(in the category of dg categories), so after all, the definition above does not depend on a
basis for I.
Example A.11 (curved type D modules over dg I-algebras). We start with the same
category C as in the previous example, but we fix a central element ac ∈ Z(A), the curvature,
and define the category of curved type D modules with curvature ac as Cxac(Mat(C)). For
a more explicit, but less concise definition, see definition III.1.4.
Remark A.12. In the Heegaard Floer community, the term “curved” seems to be the
accepted attribute for algebraic structures for which some differential is non-vanishing; how-
ever, the first written reference (that I am aware of) in which this terminology is used is of
very recent date [Zem16].
Example A.13 (type A structures over an A∞-algebra over I). Let A be an A∞-algebra
over a ring of idempotents I over F2. As in example A.9, fix a basis {ik}k∈I of idempotents
of I, where I is some index set. Let C be the category with one object for each basis element
in I, just as for type D structures. However, a morphism in a hom-object Mor(i1, i2) of C is
given by a sequence of vector space homomorphisms
(fi : i1.A⊗(i−1).i2 → F2)i≥1
where composition is defined by
(f ◦ g)i :=
∑
j+k=i+1
fj ◦ (gk ⊗ idA⊗(j−1)).
The differential ∂ is given by
(∂(f))i(a1 ⊗ · · · ⊗ ai−1) :=
∑
j+k=i+1
i−k∑
l=1
fj(a1 ⊗ · · · ⊗ µk(al ⊗ · · · ⊗ al+k−1)⊗ · · · ⊗ ai−1).
We define the category of type A structures by Cx0(Mat(C)). We define the category of
strictly unital type A structures by restricting to those objects (O, dO) such that
dO(·, 1) = idO
and
dO(·, a1 ⊗ · · · ⊗ ai−1) = 0 if i > 2 and aj = 1 for some j = 1, . . . , i− 1
and morphisms to those satisfying
f(·, a1 ⊗ · · · ⊗ ai−1) = 0 if i > 1 and aj = 1 for some j = 1, . . . , i− 1.
We say a type A structure (O, dO) is bounded if (dO)i = 0 for sufficiently large i.
A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES 101
Remark A.14. When we describe type A structures as directed graphs, it is useful to
fix a basis of the algebra A. Then, we label an arrow corresponding to a morphism f by the
formal sum of those tuples/tensor products of basis elements of the algebra A on which f is
non-zero. In this language, composition of two morphisms f and g can be described as the
sum of all concatenations of labels for f and g (modulo 2).
To describe the differential in these terms, we introduce the following notation. For a tuple
of basic algebra elements a = (a1, . . . , ai−1), define
Mi(a) :=
∑
j+k=i+1
i−k∑
l=1
(a1, . . . , µk(al ⊗ · · · ⊗ al+k−1), . . . , ai−1).
Now consider a morphism fa whose only label is a. Then the arrow of ∂(fa) is labelled by
all tuples of basis algebra elements b = (b1, . . . , bj−1) for which the a-component of Mj(b)
is 1.
Example A.15 (type AA bimodules). Let A and B be two A∞-algebra over rings of
idempotents I and J over F2, respectively. Fix a basis {ik}k∈I of I and {jl}l∈J of J , where
I and J are some index sets. Let the objects in C be of the form (ik, jl) for some (i, j) ∈ I×J .
A morphism in Mor((ik, jl), (ik′ , jl′)) is given by a sequence of vector space homomorphisms
(fm,n : ik′ .A⊗(m−1).ik × jl.B⊗(n−1).jl′ → F2)m,n≥1
where composition is given by
(f ◦ g)m,n :=
∑
i+k=m+1
j+l=n+1
fi,j ◦ (idA⊗(i−1) ⊗gk,l ⊗ idB⊗(j−1)).
For a = a1 ⊗ · · · ⊗ am−1 and b = b1 ⊗ · · · ⊗ bm−1, the differential ∂ is given by
(∂(f))m,n(a, b) :=
∑
j+k=m+1
i−k∑
l=1
fj,n(a1 ⊗ · · · ⊗ µk(al ⊗ · · · ⊗ al+k−1)⊗ · · · ⊗ ai−1, b)
+
∑
j+k=n+1
i−k∑
l=1
fm,j(a, b1 ⊗ · · · ⊗ µk(bl ⊗ · · · ⊗ bl+k−1)⊗ · · · ⊗ bi−1).
We define the category of type AA A-B-bimodules by Cx0(Mat(C)). We define the category
of strictly unital type AA structures by restricting to those objects (O, dO) such that
dO(·, 1) = idO = dO(1, ·)
and
dO(a1 ⊗ · · · ⊗ ai−1, ·, b1 ⊗ · · · ⊗ bj−1) = 0 if i+ j > 3 and (aj = 1 or bj = 1) for some j
and morphisms to those satisfying
f(a1 ⊗ · · · ⊗ ai−1, ·, b1 ⊗ · · · ⊗ bj−1) = 0 if i+ j > 2 and (aj = 1 or bj = 1) for some j.
As in remark A.6, we can easily translate all of this into the language of labelled graphs
after fixing a basis of A and B.
102 A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES
Remark A.16. In the proofs of the glueing results in section III.2 and also in section III.4,
we sometimes need to change the underlying algebra of the algebraic structures that we are
working with by some algebra homomorphism pi : A→ B (which, in most cases, is a quotient
map). In terms of the categorical description above, such a homomorphism corresponds to
a functor
F˜pi : CA → CB,
where CA and CB are the two categories corresponding to the algebras A and B. If pi respects
the differentials on A and B, then so does F˜pi. Then F˜pi in turn induces a functor
Fpi : Cx∗(Mat(C))→ Cx∗(Mat(D))
that likewise respects the differentials on both sides. In particular it sends homotopic com-
plexes to homotopic ones. Also note that a curved type D structure might become an
ordinary type D structure, if pi(atw) = 0.
Definition A.17 (pairing type D and type A structures). Let M be a type A structure
and N a type D structure over the same dg algebra A over a ring I of idempotents, together
with a fixed basis of I and A. We now reformulate the definition of the chain complex
(M  N, ∂) from [Zar09, definition 7.4] and [LOT08, section 2.4] in terms of the graphs
associated to M and N . The generators of M  N are defined by pairs of vertices in M
and N labelled by the same idempotents. Given two such pairs (v1, w1) and (v2, w2), the
(v2, w2)-component of ∂((v1, w1)) is equal to the number of pairs (s1, s2) (modulo 2), where
s2 is a sequence of labels of consecutive arrows along a path from w1 to w2 in N , s1 is a
label on an arrow from v1 to v2 and s1 = s2.
If we start with a type AA module and a type D module, the pairing is defined in the same
way, except that we compare the sequences of labels in the type D structure only to one
component of the labels of the type AA structure and record the other component in the
output, which is a type A structure.
Example A.18 (twisted complexes). Let (C, µ) be an A∞-category. Then we can define
Mat(C) just as in A.4, with higher multiplications defined in the obvious way. We can also
generalise the definition of Cx0(C) from A.5 to the A∞-setting by modifying the differential
D such that we use all A∞-operations,
D(f) :=
∑
k,l≥0
µk+l+1(dO′ , . . . , dO′︸ ︷︷ ︸
k
, f, dO, . . . , dO︸ ︷︷ ︸
l
),
and similarly generalise higher multiplications, while restricting to upper triangular precom-
plexes to make sure that all sums are finite. We denote the A∞-category Cx0(Mat(C)) by
TwC and call it the category of twisted complexes over C. The differential D above is usually
denoted by µtw1 , multiplication by µtw2 , and higher multiplications by µtwi , i > 2; for more
details, see [Sei08, section I.3l]. In this setting, the cancellation lemma and clean-up lemma
only hold under certain assumptions that, again, are needed to ensure that all sums are
finite.
A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES 103
We now state and prove the two central lemmas mentioned in the introduction.
Lemma A.19 (Cancellation Lemma). Let (X, δ) be an object of Cx∗(Mat(C)) for some
differential graded category C and suppose it has the form
(Z, ζ)
(Y1, ε1) (Y2, ε2)
a
c
f
b
e
d
where (Y1, ε1), (Y2, ε2), (Z, ζ) ∈ objCxpre(Mat(C)), f is an isomorphism with inverse g.
Then (X, δ) is chain homotopic to (Z, ζ + bgc).
Remark A.20. We usually apply this lemma to the case where (Y1, ε1) = (Y2, ε2) and
f is the identity map.
Proof. First of all, let us check that (Z, ζ + bgc) is indeed an object of Cx∗(Mat(C)):
(ζ + bgc)2 + ∂(ζ + bgc) = ζζ + ζbgc+ bgcζ + bgcbgc+ ∂(ζ) + ∂(b)gc+ b∂(g)c+ bg∂(c)
= ζζ + (ζb+ ∂(b))gc+ bg(cζ + ∂(c)) + ∂(ζ) + b∂(g)c+ bgcbgc
= ζζ + (bε1 + df)gc+ bg(ε2c+ fa) + ∂(ζ) + b∂(g)c+ bgcbgc
= ζζ + dc+ ba+ ∂(ζ) +((((
((((b (D g) + gcbg) c = ∗.
For the last step, we observe that
gcbg = gD(f)g = D(gfg) +D(g)fg + gfD(g) = D(g).
Next, we define two chain maps
F : (Z, ζ + bgc)→ (X, δ) and G : (X, δ)→ (Z, ζ + bgc)
by
(Z, ζ + bgc) (Z, ζ)
(Y2, ε2)
(Y1, ε1)
1
gc
a
c
e
d
f
b and
(Z, ζ) (Z, ζ + bgc)
(Y2, ε2)
(Y1, ε1)
a
c
1
e
d
bg
f
b
respectively. One easily checks that indeed D(F ) = 0 and D(G) = 0. Indeed, the only
non-trivial terms we need to compute are
gc(ζ + bgc) + ε1gc+ ∂(gc) + a = g(cζ + ∂(c)) + (ε1g + ∂(g))c+ a+ gcbgc
= g(fa+ ε2c) + (ε1g + ∂(g))c+ a+ gcbgc
= (D(g) + gcbg) c = 0
104 A. ALGEBRAIC STRUCTURES FROM DG CATEGORIES
for the first identity and similarly
(ζ + bgc)bg + bgε2 + ∂(bg) + d = (ζb+ ∂(b))g + b(gε2 + ∂(g)) + d+ bgcbg
= (df + bε1)g + b(gε2 + ∂(g)) + d+ bgcbg
= b (D(g) + gcbg) = 0.
for the second identity. Now, GF = 1Z and conversely, it is not hard to check that
FG = 1Z +D(H),
where H is the homotopy given by the dashed line in the following diagram:
(Z, ζ) (Z, ζ + bgc) (Z, ζ)
(Y2, ε2) (Y2, ε2)
(Y1, ε1) (Y1, ε1)
a
c
1 1
gc
a
c
e
d
bg
g
e
d
f
b
f
b

Lemma A.21 (Clean-up Lemma). Let (O, dO) be an object in Cx∗(C) for some differential
graded category C. Then for any morphism h ∈ Mor0((O, dO), (O, dO)) for which
h2, hD(h) and D(h)h
vanish, (O, dO) is chain homotopic to (O, dO +D(h)).
Proof. We can easily check that (O, dO +D(h)) is an object in Cx∗(Mat(C)):
(dO +D(h))
2 + ∂(dO +D(h)) =
(
d2O + ∂(dO)
)
+
(
dOD(h) +D(h)dO + ∂(D(h))
)
+D(h)D(h)
The first term on the right gives (∗) and the last term vanishes, which can be seen by
applying the differential D to hD(h) = 0. The middle term also vanishes, which can be seen
by expanding D(h) = dOh+ hdO + ∂(h) and using the fact that a term (∗) commutes with
any morphism. The chain isomorphisms between the two objects are given by
(O, dO) (O, dO +D(h))
1+h and (O, dO +D(h)) (O, dO).
1+h
Indeed, these two morphisms lie in the kernel of D, since hD(h) and D(h)h vanish. Their
composition is equal to 1 + h2 = 1. 
APPENDIX B
Proof of the generalised clock theorem
First of all, we introduce some terminology, some of which is inspired by [Kau83]:
Definition B.1. A D-graph is a planar graph embedded in the closed disc D2 such
that at least one vertex lies on ∂D2. A universe is a D-graph such that all vertices in the
interior of D2 are 4-valent and the ones on ∂D2 are 1-valent. A face of a graph embedded
into D2 is a connected component of the complement of this graph in D2.
Remark B.2. We define generalised Kauffman states of universes just as for tangles. In
fact, a universe is obtained from a tangle diagram by forgetting the under/over information
at each crossing, and conversely, any universe gives rise to some tangle diagram.
Definition B.3. A clocked state of a tangle/universe is a generalised Kauffman state
with the property that one cannot perform any anticlockwise transposition move.
Proposition B.4. In any universe one cannot perform an infinite number of clockwise
transposition moves in sequence.
Proposition B.5. The set of Kauffman states of a fixed site of a universe has a unique
clocked state if it is not empty.
Proof of the generalised clock theorem I.1.17. If the set of Kauffman states
for a given site is empty, there is nothing to show. Otherwise, given any two Kauffman states
x and x′ of a fixed site, proposition B.4 gives us two clocked states y and y′ with x ≤ y and
x′ ≤ y′. Proposition B.5 tells us that y = y′, so the set of Kauffman states of a fixed site is
a (finite) join-semilattice.
The duals of propositions B.4 and B.5 follow from considering mirror diagrams and observing
that the mirror of a clockwise transposition move is a anticlockwise transposition move and
vice versa. Thus we see that the set of Kauffman states of a fixed site is also a (finite)
meet-semilattice and the result follows. 
Proof of proposition B.4. The argument from [GL86, lemma 4] carries over to the
tangle case; for completeness, we recall it here: First of all, note that transposition moves do
not change sites. Hence, the marker of an outermost vertex, i. e. one which meets an open
unoccupied region, cannot make a complete cycle. A marker at a vertex which has a common
edge with an outermost vertex cannot make two cycles because in the course of each cycle
it has to interact with the marker at the outermost vertex and that one cannot make a full
cycle. Similarly, a marker at a vertex which is n = 2 edges away from an outermost vertex
cannot make 2n = 22 = 4 full cycles, and so on. 
For proposition B.5, we first generalise the correspondence used in [GL86] between span-
ning trees and states, see also [Kau83, theorem 2.4].
105
106 B. PROOF OF THE GENERALISED CLOCK THEOREM
Given a connected oriented universe U , partition the set of faces of U into two subsets by
shading each component which is separated from an arbitrarily fixed region by an odd num-
ber of edges. Form a new D-graph G(U) = G with one vertex for each shaded region and
one edge for each crossing shared by these shaded regions, such that the vertices lie in their
corresponding regions and those vertices of all open regions lie on ∂D2. In a similar way, we
obtain a D-graph H from the unshaded regions. Note that H is the dual graph of G.
Definition B.6. Given a D-graph G and its dual G∗, a D-forest F is the union of a
maximal forest FG in G and its dual F ∗G in G
∗ such that each forest has at least one point
on ∂D2, together with a specification of a root on ∂D2 for each tree in F .
Lemma B.7. There is a one-to-one correspondence between D-forests in G(U) and Kauff-
man states of a (connected) universe U .
Proof. The argument from [Kau83, lemma 2.4] carries over, but we spell this out
explicitly nonetheless. Given a D-forest F , we orient its edges in the canonical way, that is
arrows point away from the roots. For each edge of F , place a marker into the region that
the arrow of the edge is pointing towards. We claim that this defines a valid Kauffman state.
Indeed, for each vertex in F except the roots there is exactly one arrow pointing towards
this vertex. Since vertices in F correspond one-to-one to faces of U , we are done.
Going from Kauffman states back to forests in G is now straightforward: For each marker at
a crossing, we draw an edge between the corresponding vertices, according to the rule used
above. Note that at each crossing, there is exactly one edge, so we get two disjoint subgraphs,
one in G(U) and one in G(U)∗. There is no simple cycle in either of these subgraphs.
Otherwise, there would be a subdiagram D′ traced out by the cycle in D2. A simple Euler
characteristic argument leads to a contradiction: Say, there are n vertices in this cycle (each
of which becomes a 2-valent vertex in D′) and m crossings of U in D′. Then we have a total
of (m + n) vertices, 2(m + n) edges and therefore 1 + 2(m + n) − (m + n) = (m + n + 1)
faces in D′. Only the inner faces need to be occupied by markers, which means m markers
have to occupy (m+ 1) faces. Contradiction.
Finally, every unoccupied region of U becomes a root in F . It is clear that every tree in F
has exactly one root. 
Next, we need to describe what a transposition move looks like in the language of D-
forests. The following picture illustrates the clockwise transposition move. The dotted edges
denote those edges in the graph G and its dual G∗ that do not belong to the forest. The
dashed edges denote edges in the graph that may or may not be in the forest.
b
b
−→ b
b
−→
B. PROOF OF THE GENERALISED CLOCK THEOREM 107
We are now ready to give an outline of the proof of proposition B.5. The proof goes by
induction on the number of crossings in a tangle diagram. The induction step relies on the
fact that a certain edge of the graph G belongs to any clocked forest. This is the content of
the next proposition.
Proposition B.8. For any site of a (connected) universe, there exists a pair of two
adjacent roots with the following property: For all r ≥ 0, the number of roots in the next 2r
boundary faces (moving anticlockwise along the boundary, that is to the right of the picture
below, see also figure 46) contain at least r roots.
Furthermore, suppose we have a site of a connected universe together with two such roots.
Since the diagram is connected, we can consider the first crossing that one reaches along their
common edge from the boundary of the disc. Suppose the two other regions at this crossing
are no roots, as in the following picture.
root root
no root no root
∂D2
Then the marker of the crossing in the picture above sits in the upper left region for any
clocked Kauffman state of this universe with respect to the fixed site.
We postpone the proof of proposition B.8 and first see how it implies proposition B.5.
Proof of proposition B.5. The start of the induction is given by all those tangle
diagrams that have exactly one Kauffman state. For these, the statement is trivially true.
For the induction step, consider a universe U and a clocked Kaufman state x of U . Without
loss of generality, we may assume that the diagram is connected. Indeed: Otherwise we
can consider its connected components by splitting the diagram at a region with more than
one component on ∂D2 and repeating this process as often as necessary. Observe that the
restrictions xi of x to the connected diagrams are also clocked and that the sites of the
restrictions of any other (clocked) Kauffman state of the same site as x agree with the sites
of the xi. Hence, if we can show the proposition for the xi, then it also holds for x.
Now, consider two adjacent roots and the first crossing that one reaches along their common
edge from the boundary of the disc. Obviously, not all four regions of this crossing can be
roots. If there is exactly one additional root at this crossing, one can split the diagram into
two parts like so:
b
root root
root∂D
2
∂D2
−→
root
root
root
root
new root
Note that the marker at the crossing has to be where it is for any Kauffman state. Also
note that this splitting reduces the number of crossings by 1, so we can apply the induction
hypothesis to both diagrams and we are done. So without loss of generality, we may now
also assume that all adjacent roots locally look as in proposition B.8.
108 B. PROOF OF THE GENERALISED CLOCK THEOREM
We consider the subdiagram obtained by removing a small neighbourhood of the edge from
the boundary to this crossing. The clocked Kauffman state x restricts to a clocked Kauffman
state x| of the subdiagram.
root root
b
∂D2
The subdiagram has one crossing fewer and we can apply the induction hypothesis, i. e. x|
is the unique clocked Kauffman state of the subdiagram with respect to the site of x. Since
this site is uniquely determined by the site of x and the marker in the picture above, x is
the unique clocked Kauffman state of the universe U with respect to the site of x. 
Proof of proposition B.8. The existence of a pair of roots with the special property
required above follows from a standard argument: Consider the 2n line segments on ∂D2,
choose one as a starting point and walk along ∂D2 in anticlockwise direction. Define a
function from the set of segments to Z by adding +1 for each root segment and −1 for
an occupied segment, see the illustration below. The value of the function at the very
last segment will be 2. Then consider the rightmost segment where the function takes its
minimum. This will be a non-root, followed by two successive roots. By construction, these
have the required property.
b
b
b
root pair
1 2n
segments on ∂D2
Let us denote the vertex corresponding to the upper right (resp. left) region of the picture
in proposition B.8 by p0 (resp. p1). Suppose the edge from the right root to p1 does not
belong to the D-forest F corresponding to a clocked Kauffman state. We want to show that
somewhere in the diagram, we can perform a anticlockwise transposition move, so x cannot
be a clocked state. Let us call its dual (which is in the D-forest) e0. Then e0 points away
from the left root:
root root
b b
bb
p1 p0
e1
e0
∂D2
Note that p1 has (exactly) one incoming edge e1 of F , since p1 is not a root. Consider the
edges in F that the dual e∗0 of e0 meets on its way to e1 during a anticlockwise rotation
B. PROOF OF THE GENERALISED CLOCK THEOREM 109
around p1, see the picture below. (Note that there might be no such edge.)
bp1
e1
e∗0
Let F ′ be the tree consisting of p1, these edges and all vertices and edges in F that can
be reached from these edges when following the arrows. Then the special property of our
chosen pair of adjacent edges enables us to prove the following:
Lemma B.9. Any vertex q that belongs to F ′ does not lie on ∂D2.
Proof. Suppose there is such a vertex q. Note that p1 = q is also allowed. We have the
following situation:
root root
m interior crossings
︸ ︷︷ ︸
2r endpoints︸ ︷︷ ︸
2r boundary regions
bb
bp1
b
q
b
b
s crossings between p and q
∂D2
Figure 46. The Euler characteristic argument for the proof of lemma B.9
We compute the number of faces of the disc enclosed by the grey curve, using an Euler
characteristic argument. There are m interior crossings, say, 2r endpoints on the boundary
between q and the right root, and s crossings between p1 and q. (Again, s = 0 is allowed.)
Then we have
#{vertices} = 2r + s+ 2 +m
#{edges} = (2r − 1) + s+ 3 + 12(4m+ 2r + 2s+ 2)
= 3r + 2s+ 3 + 2m
⇒ #{faces} = 1 + #{edges} −#{vertices} = r + s+ 2 +m
Hence the number of occupied faces is
#{faces} − (s+ 1)− (≥ r + 1) ≤ m.
110 B. PROOF OF THE GENERALISED CLOCK THEOREM
However,
#{markers} = m+ 1.
Contradiction! 
Corollary B.10. The vertex p2 that we reach by following the dual edge of e1 to the
left of e1 is in the full subtree starting at p0, see figure 47.
Proof. Choose a small contractible neighbourhood of the tree F ′. Then its boundary
gives us a path in D2 from the region corresponding to the vertex p0 to the region containing
p2 that is disjoint from the D-forest F . 
b
b
p0p1
p2
e0
e1
e2
Figure 47. An illustration of corollary B.10
We can now finish our proof of proposition B.8. The corollary above tells us in particular
that p2 is not a root, so there is exactly one incoming edge e2. Note that if p0 = p2 (or
equivalently e0 = e2), we are done, because we can perform a anticlockwise transposition
move. If p0 6= p2, we can repeat the argument for p2 in place of p1. It is now clear that p3 is
in the subtree starting at p1 and therefore has an incoming edge. This is where we needed
the lemma in the first iteration step.
This algorithm terminates because there are only finitely many vertices in the shaded region
in figure 47 and the number of vertices in the corresponding shaded area in each iteration
step strictly decreases unless the algorithm terminates during this step. 
APPENDIX C
Manual for APT.m
version 1.1, written in Mathematica 10.3.1.0 for Linux x86 (64-bit)
The Mathematica package [APT.m] provides a simple tool for calculating the polynomial
tangle invariants ∇sT from definition I.1.8. It can also be used to compute the generators of
ĈFT from the “standard” Heegaard diagrams in example II.2.5. This manual should be read
alongside the notebook [APT.nb]. APT stands for “Alexander Polynomial for Tangles”.
Input preparation. We start by loading the package [APT.m] as follows:
<< APT.m
Given an oriented 2n-ended tangle T , we choose a connected tangle diagram with at least
one crossing. To translate this data into computer-speak, we enumerate all crossings and
give a unique name to each region. Usually, we use letters a, b, c,. . . for this and label the
open regions first, in anticlockwise order, as illustrated in the example in figure 48. Finally,
we choose colours of the tangle components, for which we usually use the letters p, q, r,. . .
1
2
3
5
4
p
q
a
b
c
d
e f
g
h
Figure 48. A labelling of regions and crossings of a tangle diagram
Input data. The data for a tangle diagram is entered into a Mathematica notebook as
a list
v={v0,v1,v2,...,vm},
where v0 is a list of open regions of the tangle (in anticlockwise order), m is the number of
crossings and, for 1 ≤ i ≤ m, vi is a list with the following seven entries:
• The first four entries (vi)1 to (vi)4 are the labels of the regions in the four quadrants
of the ith crossing. The first entry is the label of the region between the two
outward-pointing arrows and then we go in anticlockwise direction, as illustrated
in figure 49a.
111
112 C. MANUAL FOR APT.M
(vi)1
(vi)2
(vi)3
(vi)4
(a) The ith crossing
ou
ou−1
h−1o−1u−1
o−1u
δ
1
2
δ
1
2
(b) A positive crossing (L)
o−1u−1
ou−1
hou
o−1u
δ−
1
2
δ−
1
2
(c) A negative crossing (R)
Figure 49. (a) shows the order of the four quadrants vi at an oriented crossing.
(b) and (c) show the Alexander codes used in [APT.m]. They coincide with those
from figure 6 after substituting o 7→ o1/2 and u 7→ u1/2. The over-strand is coloured
by o and the under-strand by u.
• The fifth entry (vi)5 is either L or R, depending on whether i is the label of a
positive (left) or negative (right) crossing, see figure 49b and 49c.
• The entries (vi)6 and (vi)7 are the colours of the over- and under-strands, respec-
tively.
It is not hard to see that the tangle is uniquely determined by v.
Example C.1. Let T be the tangle diagram shown in figure 48. Then the list v = v(T )
is given by
v = {{a,b,c,d},
{h,f,b,c,R,q,q},
{g,f,h,c,R,q,q},
{d,f,g,c,R,q,q},
{a,e,f,d,L,p,q},
{f,e,a,b,L,q,p}}.
Calculation of ∇sT . To explain the main function AlexpolyGdh[v] of this package,
we introduce the following notation: As in definition I.1.8, let c(x) be the labelling of a
Kauffman state x using the Alexander codes from figure 49. For a site s of T , let s(s) be
the word obtained by concatenating all labels of regions in s in alphabetical order; we set
s(∅) = 1. Similarly, let g(x) be the concatenation of all labels of regions occupied by x in
the order of the crossings. Then AlexpolyGdh[v] computes∑
s∈S(T )
x∈K(T,s)
c(x) · s(s) · g(x).
The terms g(x) uniquely determine the Kauffman states and thus the generators of the
tangle Floer complex ĈFT from the “standard” Heegaard diagrams in example II.2.5.
There is also a function AlexpolyPdh[v], which is the same as AlexpolyGdh[v] except that
the terms g(s) are omitted. There are six other variants of AlexpolyPdh and AlexpolyGdh
which “forget” the homological grading and/or the δ-grading by setting h = −1 and/or
δ = 1. The names of the functions are obtained from AlexpolyPdh and AlexpolyGdh simply
by dropping the letters h and/or d, respectively. In particular, the polynomial invariants ∇sT
are equal to the coefficients of s(s) in AlexpolyP[v] – up to substitution of all colours of T
by their square roots, see figure 49.
C. MANUAL FOR APT.M 113
d hgdfe h-1 d hgfde d h hfgde d fhgde h2
d hgdef
Figure 50. The output of SimpleGrid applied to the polynomial AlexpolyGh[v]
from example C.2, after setting the variables a, b and c equal to 0. The heavily
shaded entry has Alexander grading p0q0. The lightly shaded entries indicate the
axes p0qn (horizontal, pointing right) and pnq0 (vertical, pointing down).
Example C.2. In the example from above, we obtain
AlexpolyPdh[v] = d δ−1q−6p−2h−1 + b p2δ−1q−6 + d p2δ−1q−6 + d δ−1q−2p−2
+ b hp2δ−1q−2 + d hq2δ−1p−2 + b h2p2q2δ−1 + b h2q6δ−1p−2
+ d h2q6δ−1p−2 + b h3p2q6δ−1 + c p−2δ−
1
2 + c hp2δ−
1
2
+ a q−6h−1δ−
1
2 + c q−4p−2h−1δ−
1
2 + c p2q−4δ−
1
2 + 2a q−2δ−
1
2
+ 2a hq2δ−
1
2 + c hq4p−2δ−
1
2 + c h2p2q4δ−
1
2 + a h2q6δ−
1
2
and
AlexpolyGh[v] = c hcgfe p−2 + c h hcgef p2 + a hgfea q−6h−1 + d hgdfe q−6p−2h−1
+ d hgdef p2q−6 + b hgfeb p2q−6 + c hgcfe q−4p−2h−1
+ c hgcef p2q−4 + a hfgea q−2 + a hgfae q−2 + d hgfde q−2p−2
+ b h hfgeb p2q−2 + a fhgea hq2 + a h hfgae q2 + d h hfgde q2p−2
+ b fhgeb h2p2q2 + c chgfe hq4p−2 + c chgef h2p2q4 + a fhgae h2q6
+ b bhgfe h2q6p−2 + d fhgde h2q6p−2 + b bhgef h3p2q6
Basic graphical output. As example C.2 illustrates, it can be quite hard to understand
polynomials with many terms just from their algebraic expressions. Fortunately, the package
[APT.m] also provides some graphical tools for dealing with this.
SimpleGrid[poly] rearranges the monomials of a Laurent polynomial poly in a grid
with two axes, one for the colour p and one for q; see figure 50 for an example.
AlexpolytableGdh[v,poly] is based on SimpleGrid[poly]. Its input is a tangle
v and a Laurent polynomial poly which is usually the output of the function
AlexpolyGdh or one of its variants. The output of AlexpolytableGdh is a list
of all sites s of v and the corresponding outputs of SG applied to the coefficient of
s(s) in poly.
AlexpolytablePdh[v,poly] is similar to the previous function, but it sets all
terms g(x) in poly equal to 1.
Just as for AlexpolyGdh and AlexpolyPdh, there are variants of the functions
AlexpolytableGdh and AlexpolytablePdh which “forget” the homological and/or δ-grading
in the output. Again, the names of these functions are obtained from AlexpolytableGdh
and AlexpolytablePdh by dropping the letters h and/or d.
114 C. MANUAL FOR APT.M
p
1
a
p
2
b q
3
c
q
4
d
Figure 51. The output of DConfig[v] for example C.1
More functions.
DConfig[v] graphically represents the boundary configuration of the tangle v, i. e.
the open regions of the tangle diagram and the open tangle components together
with their orientations and colours, see figure 51.
Twist[v,openregion,twist] computes the new tangle obtained from v by adding
a single crossing at the region openregion, where twist determines if the new
crossing is a positive (L) or a negative (R) crossing. (Note: This function only
works correctly if there are fewer than 26 regions in the new diagram – otherwise
we run out of letters to label the new region with.)
Closure[v,openregion] computes the Alexander polynomial of the link obtained
from the tangle v by pairing it with a tangle of parallel strands which caps off the
region openregion.
DotGrid[v] is a function for 4-ended tangles v whose open regions are labelled (ac-
cording to our conventions) by a, b, c and d. It returns a convenient graphical out-
put, which represents Kauffman states of all sites by dots in a single 2-dimensional
grid according to their Alexander gradings, where we use the same conventions as
for the function SimpleGrid, see figure 52. The dots are labelled by their δ-grading.
They are also coloured according to the site they belong to with the following colour
conventions:
a, b, c, d.
III.0.1. Change log: versions 1.0→ 1.1.
• Changed δ-grading in Alexander codes to match new conventions.
• Defined new functions Twist, Closure, DConfig and DotGrid.
• Moved code to a Mathematica package file APT.m.
-1 -1/2 -1 -1/2 -1 -1/2 -1 -1
-1/2 -1/2 -1/2 -1/2 -1/2 -1/2
-1 -1 -1/2 -1 -1/2 -1 -1/2 -1
Figure 52. The output of DotGrid[v] for example C.1
APPENDIX D
Manual for BSFH.m
version 1.0, written in Mathematica 10.3.1.0 for Linux x86 (64-bit)
The Mathematica package [BSFH.m] provides a tool for calculating Zarev’s bordered su-
tured Floer homology for any bordered sutured manifold [Zar09]. Both type A and type D
structures can be computed with this program, as well as the various bimodule invariants.
Tools for manipulating the invariants are also provided, namely for cancellation and for per-
forming basic homotopies. This manual only covers the main functionality of the package
[BSFH.m] and should be read alongside the notebook [nb3], where we compute the bordered
sutured type A structure from remark III.2.4 and explain some more advanced features.
Input preparation. Our program [BSFH.m] implements an algorithm due to
Zarev [Zar09, theorems 7.14 and 7.15], which allows us to compute bordered sutured in-
variants combinatorially from nice Heegaard diagrams. So, given a Heegaard diagram for a
bordered sutured manifold, we first need to niceify it, using [Zar09, proposition 4.17]. How
we do this can have a significant impact on the run time of the computation for large dia-
grams, so one should keep the number of generators as low as possible. Figure 53a shows
the niceified Heegaard diagram that we use for the computation in [nb3]. We now explain
how to translate a picture like this into the input data for our program.
First, we label the intersection points of α-curves and β-curves by integers (1, 2, 3, . . . ).
Similarly, we label the regions in the Heegaard diagram that are not adjacent to any base-
points (1, 2, 3, . . . ). Furthermore, we enumerate the α-arcs and α-curves and, separately,
all β-curves. We also draw an arc diagram for each glueing surface. In [nb3], we only have
one arc diagram, which is shown in figure 53b. The orientation of the arc diagram is as
on the Heegaard surface and we draw it such that the α-arcs are to the right of the line
segments. We enumerate all the endpoints of the α-arcs from bottom to top. We also label
each component of the line segments between two endpoints by the (index of the) region
adjacent to it and its corresponding algebra element.
Basic input data. We start by loading the package [BSFH.m] as follows:
<< BSFH.m
Next, we enter the following data into the notebook. Note that for each calculation, a sepa-
rate notebook should be used.
regionsInput is a table whose rows are indexed by the regions of our Heegaard
diagram. The first entry of the ith row is equal to i, then follow the vertices of
that region. These are ordered by the boundary orientation of the region, i. e.
115
116 D. MANUAL FOR BSFH.M
ρ τ
δε
b
1
b
2
b
3
b
4
b
5
b
6
b
7
b
8
b
9
b
10
b
11
b
12
b
13
b
14
b
15
b
16
b
17
1(τ1)2(τ2)
3(τ3)
4(δ1)
5(δ2) 6(δ3)
7
8 9
10
1112
13
14
α1
(d)
α2(a)
α3(b)
α4
α5
β1 β2
β3
β4
(a) A nice Heegaard diagram with many labels
1
2
3
4
5
6
7
8
9
10
δ1
δ2
δ3
τ1
τ2
τ3
4
5
6
1
2
3
α1(d)
α2(a)
α3(b)
α4
α5
(b) An arc diagram for the
Heegaard diagram on the
left
Figure 53. The starting point of any calculation is a nice Heegaard diagram and
an arc diagram. Here, we have drawn them for the computation in [nb3].
anticlockwise, starting with a vertex which is the start of a segment of an α-curve
or α-arc in the boundary of the region. For bigons, the last two entries remain
empty, i. e. occupied by the symbol .
alphaarcsInput is a table whose rows are indexed by the α-arcs. Again, the first
entry should be the index of the arc; the second is a list of (the indices of) all
intersection points on this arc. alphacurvesInput and betacurvesInput are the
corresponding tables for α- and β-curves.
CancellationSortListInput is a table of (indices of) intersection points with two
columns, so each row corresponds to a pair of intersection points. This table has
an effect on the order of the generators and is important for the initial cancellation
after computing the invariant.
Usually, we start with a Heegaard diagram which is not necessarily nice and then
niceify it using finger moves, thereby creating lots of new intersection points and
generators. In the initial cancellation step, the program attempts to cancel such
generators, thereby reversing the effect of niceification. The order in which this is
done is determined by the order of the generators, and this, in turn, is determined
by CancellationSortListInput.
So the pairs of intersection points should be those that can be removed by a reversed
finger move. The order of the intersection points for each pair is the same as for
regionsInput, using the bigon which connects these two points and which is re-
moved by the reversed finger move. This bigon accounts for the identity component
in the differential which the program will attempt to cancel.
RelevantGeneratorQ[v] is a function used in the generation of generators. It takes
a generator v of the Heegaard diagram and decides if it should be used in the
D. MANUAL FOR BSFH.M 117
calculation of the invariant. For type A structures or bimodules, it can make sense
to forget some generators in certain idempotents (see for example theorem II.3.7),
but this functionality should be used with care. As a default, set equal to True.
RelevantGeneratorGenerationQ[v] also plays an important role in the generation
of generators. It takes a list of α-curves and decides if generators occupying these
curves should be used in the calculation of the invariant. This is a quick check used
for speeding up the generator generation. Like RelevantGeneratorQ, it should be
used with care. As a default, set equal to True.
S1BoundaryInput is used for the calculation of the bordered sutured algebra: It is
a list of (indices of) regions for each line segment in the arc diagram for the first
glueing surface S1, ordered from bottom to top. If S1 plays the role of a type A
side, this ordering is reversed internally, compare with figure 54. This is opposite
to Zarev’s conventions. Similarly for S2BoundaryInput.
TypeS1 should be set to either TypeA or TypeD, depending on which structure we
would like to compute for the first glueing surface S1. Similar for S2; if there is no
second glueing surface S2, either inputs are fine.
ArcDiagram1MatchingsInput is a list of the lists of (the indices of) endpoints of the
α-arcs in the arc diagram for S1, ordered by the indices of the α-arcs. Each entry
itself should be ordered. Same for S2.
S1AlwaysOccupiedAlphaArcs is a list of indices of α-arcs for S1 that are oc-
cupied by all generators in the set of relevant generators determined by
RelevantGeneratorQ[v] above. Similarly, S1NeverOccupiedAlphaArcs is a list
of indices of α-arcs for S1 that are cannot be occupied by any relevant generator.
As a default, set both equal to {}. S1PotentiallyOccupiedAlphaArcs is the list
of remaining indices of α-arcs for S1. Same for S2.
NS1oaa is equal to the total number of occupied α-arcs for S1. Same for S2.
NR and NI are the total number of regions and intersection points, respectively.
Running a calculation. We run the program in several separate steps and substeps
by evaluating the variables called
Step<i><name of step i> and Step<i>x<j><name of step i>.
They are modules defined in the package [BSFH.m] which group together blocks of code
that compute the various components of the invariant, like for example the generators, the
domains, parts of the algebra, etc.
Step 0: Double-checking basic input data. Before we start the actual calculation,
we make sure that the input data that we have entered so far is consistent and does indeed
correspond to our Heegaard diagram. At the end of the cell containing the basic input
data above, we call Step0Preparation. This returns a list of results of automated sanity
checks and some additional outputs, which one should check manually. Perhaps the most
useful of these are the lists named 0 vertices, +1 vertices, etc., which partition the set
of intersection points into five subsets as follows. An intersection point is in <n> vertices
if the sum of unlabelled regions (i. e. those adjacent to basepoints) in the four quadrants
118 D. MANUAL FOR BSFH.M
b
x
b
y1
2
+1
(a) A boundary domain
x 7→ ιx.a.ιy ⊗ y
ιx a ιy
1
2
(b) The type D side
(x, ιx.a.ιy) 7→ y
ιyaιx
1
2
(c) The type A side
Figure 54. Our conventions for counting domains and the algebras of moving
strands for type D and type A sides. For type A sides, the program internally
reverses the labelling of the endpoints of the α-arcs. This is opposite to Zarev’s
conventions.
around it equals n, where each region contributes as follows: Walking around an intersection
point in anticlockwise direction, an unlabelled region that we enter through a β-curve counts
as +1, one that we exit through a β-curve as −1.
Backups. At various stages throughout the program, backups of preliminary results are
made. For this, we need to specify a path to the directory where the backups should be
stored. They can then be recalled for example like this:
<< (BackupFilePath <> "_BeforeCancellation.mx");
Step 1: The algebra. In the first step of the calculation, we compute the glueing
algebra. This is done in substeps Step1x1Algebra to Step1x5Algebra. After each of the
first two substep, a few more inputs are needed. Step1x1Algebra returns a list of idem-
potents for each glueing surface. We can then choose names for these in S1IdemNames and
S2IdemNames.
Furthermore, after evaluating Step1x2Algebra in [nb3], there two inputs called
S1SubalgebraGensInput and S2SubalgebraGensInput. Here, we can specify names of gen-
erators for a suitable subalgebra of each glueing algebra. The point of this is the following:
algebra elements are internally represented by integers. If a generator lies in the specified
subalgebra, it will later be represented in any graphical output by the corresponding product
of names for the generators of the subalgebra. As a default, one might want to give a name
to each algebra element with a single moving strand between consecutive starting and ending
points. Also, if this functionality is not required, one can simply set both variables equal
to {}.
S1SubalgebraGensInput and S2SubalgebraGensInput are two matrices where each row
corresponds to a generator of the respective subalgebra. The first entry is a string that
represents this generator. The second entry is a LATEX-friendly version of this string. The
third and fifth entries are the starting and ending idempotent indices; the fourth a list of
start- and endpoints of the moving strands.
After this, we evaluate the remaining substeps Step1x3Algebra to Step1x5Algebra.
Steps 2–3: Generators and domains. In step Step2Generators and substeps
Step3x1Domains to Step3x4Domains, we compute the generators and the structure map
of the invariant, respectively. Our conventions for counting domains are shown in figure 54.
Some of the steps offer intermediate results which can be used as additional sanity checks.
D. MANUAL FOR BSFH.M 119
Basically, the invariant is fully computed after evaluating substep Step3x4Domains; the gen-
erators are stored in generators and the structure map in StructureMapsSparse. However,
usually there are quite a lot of generators, so the result at this stage is not particularly useful.
Therefore, as mentioned earlier, we offer some tools for manipulating the result.
Step 4: Cancellation. Step4CancellationPreparation provides the tools for cancel-
lation. For the initial cancellation, one should always use
CancellationSparse[StructureMapsSparse, generators].
The output of this function is a list
{<cancelled structure map>,<remaining generators>}.
The number of remaining generators should be the same as for the initial non-niceified
(admissible) Heegaard diagram – provided we have set up CancellationSortListInput
correctly. Note that CancellationSparse uses sparse arrays for the structure maps, but
converts them into their normal form for the output.
One might want to perform some further cancellation. There are two tools for this:
Cancellation[StructureMaps, gens, cancel] works very similar to the function
CancellationSparse. gens is the set of generators underlying StructureMaps.
cancel is an optional third argument where one can specify a list of generator pairs
{<end of cancelling arrow>,<start of cancelling arrow>}.
The iteration will stop at the first instance the conditions for cancellation are not
satisfied, or if it has run through the complete list. If the third argument is missing,
any identity morphism will be cancelled (if possible). In this case, there is a third
entry in the output, which is a list of all cancelled generator pairs.
CarefulCancellation[StructureMaps, generators] is very similar to the function
Cancellation, but it only performs those cancellations which do not result in a
complex with non-trivial maps from a generator to itself (loops) or differentials
between two generators in either directions (double arrows).
Step 5: Post-Processing. The last step Step5PostProcessing loads two main func-
tionalities of the package, namely tools for displaying the result graphically and for perform-
ing basic homotopies.
Basic graphical outputs. There are two main output formats for the whole invariant,
which are provided in Step5OutputPreparation.
PlotGraph[StructureMaps] is a very basic graphical representation. It shows a
schematic picture of the matrix StructureMaps, where each entry is represented
by a coloured box in a grid. If the colour is green, the entry is empty. If the entry
is non-empty, the colour is a shade of grey, where white represents a (relatively)
short entry, black a (relatively) long one.
ShowGraph[StructureMaps, GeneratorSet] displays the full invariant as a labelled
graph. If one only wants to see a portion of the graph, one can use the variant
ShowSubGraph[StructureMaps, GeneratorSet, Subgraph],
120 D. MANUAL FOR BSFH.M
where Subgraph is a list of (indices of) those generators forming the vertices of the
subgraph. An output of this is shown in figure 38.
There are also functions for a graphical representation of algebra elements, the most useful
one being S1H for the glueing algebra S1 and S2H for S2. Given an (index of an) algebra
element for S1, S1H returns the sum of its basic algebra elements in the (larger) moving
strands algebra. Each such element is represented by a grid with a row for each moving
strand displaying its starting point on the left and its endpoint on the right. For type A
sides, note that the indices of the start- and endpoints are reversed, see figure 54.
Performing basic homotopies. There are several functions for “adding homotopies”
in the sense of the clean-up lemma (lemma A.21). The easiest to use is the following:
AutoCleanPosns[StructureMaps, gens, pos, max] tries to simplify the matrix
StructureMaps at positions pos by a sequence of at most max homotopies. The
output is of the form
{<new structure map>,<record of added homotopies>}.
Note that this function does not always give the same output, because it uses the
pseudo-random Mathematica function RandomSample.
We also mention two other, more basic functions which are quite useful if the previous
automated function does not give quite the desired result, see for example the notebook [nb4].
Homotopy[StructureMaps, gens, homotopy, start, end] calculates the structure
map obtained by adding the homotopy to StructureMaps which is labelled by an
algebra element homotopy and goes from the generator with index start to the
generator with index end in the underlying set of generators specified by gens.
AllHomotopies[StructureMaps, pos] computes a list of all homotopies that can
remove an arrow label specified by pos in StructureMaps. Here, a homotopy has
the following form
{<algebra element>,<start>,<end>}.
Bibliography
[AAEKO] M.Abouzaid, D.Auroux, A. I. Efimov, L.Katzarkov, D. Orlov, Homological mirror symmetry for
punctured spheres, J. Amer. Math. Soc. 26 (2013), 1051-1083 (arXiv: 1103.4322)
[Ada94] C.C.Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of
Knots, W.H. Freeman & Company (1994)
[Ale28] J.W.Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928),
275–306
[Alt13] I. Altman, Introduction to sutured Floer homology , arXiv: 1304.2606v1
[Arc10] J.Archibald, The multivariable Alexander polynomial on tangles, PhD thesis (2010), University
of Toronto, available at http://www.math.toronto.edu/jfa/jana_thesis.pdf
[Aur10] D.Auroux, Fukaya categories of symmetric products and bordered Heegaard-Floer homology , J.
Gökova Geom. Topol. 4 (2010), 1–54 (arXiv: 1001.4323v3)
[Bar02] D.Bar-Natan, On Khovanov’s categorification of the Jones polynomial , Algebr. Geom. Topol. 2
(2002), 337–370 (arXiv: 0201043v3)
[Bar04] , Khovanov’s homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443–1499
(arXiv: 0410495v2)
[BL11] J.A.Baldwin, A. S. Levine, A combinatorial spanning tree model for knot Floer homology , Adv.
Math. 231 (2012), 1886–1939 (arXiv: 1105.5199v2)
[Big12] S. Bigelow, A diagrammatic Alexander invariant of tangles, arXiv: 1203.5457v1
[BCF12] S. Bigelow, A.Cattabriga, V. Florens, Alexander representation of tangles, arXiv: 1203.4590v2
[Boc07] R.Bocklandt, Noncommutative mirror symmetry for punctured surfaces, arXiv: 1111.3392v2
(brought to my attention by M.Abouzaid)
[DV16] C.Damiani, V. Florens, Alexander invariants of ribbon tangles and planar algebras,
arXiv: 1602.06191v1
[EPV15] A.P. Ellis, I Petkova, V.Vértesi: Quantum gl(1|1) and tangle Floer homology ,
arXiv: 1510.03483v1
[FJR09] S. Friedl, A. Juhász, J. A.Rasmussen, The decategorification of sutured Floer homology , J. Topol.
4 (2011), no. 2, 431–478 (arXiv: 0903.5287v4)
[GL86] P.M.Gilmer, R.A. Litherland, The duality conjecture in formal knot theory, Osaka J. Math. 23
(1986), 229–247
[HKK14] F.Haiden, L.Katzarkov, M.Kontsevich, Flat surfaces and stability structures,
arXiv: 1409.8611v2
[Han13] J.Hanselman, Bordered Heegaard Floer homology and graph manifolds, arXiv: 1310.6696
[HRW16] J.Hanselman, J.A.Rasmussen, L.Watson, Bordered Floer homology for manifolds with torus
boundary via immersed curves, arXiv: 1604.03466v1
[HW15] J.Hanselman, L.Watson, A calculus for bordered Floer homology , arxiv: 1508.05445v1
[Har83] R.Hartley, The Conway potential function for links, Comment. Math. Helv. 58 (1983), 365–378
[HHK13] M.Hedden, C.Herald, P.Kirk, The pillowcase and perturbations of traceless representations of
knot groups, Geom. Topol. 18 (2014), 211–287 (arXiv: 1301.0164v1)
[HHK15] , The pillowcase and traceless representations of knot groups II: a Lagrangian-Floer
theory in the pillowcase, arXiv: 1501.00028v1
[HR11] Y.Huang, V.G.B.Ramos, An absolute grading on Heegaard Floer homology by homotopy classes
of oriented 2-plane fields, arXiv: 1112.0290v2
121
122 BIBLIOGRAPHY
[HR12] , A topological grading on bordered Heegaard Floer homology , Quantum Topol. 6 (2015),
403–449, (arXiv: 1211.7367v2)
[Jia14] B. Jiang, On Conway’s potential function for colored links, Acta Math. Sinica (English Series)
32 (2016), no. 1, 25–39 (arXiv: 1407.3081v2)
[Juh06a] A. Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006), 1429–1457
(arXiv: 0601443v3)
[Juh06b] , Floer homology and surface decompositions, Geom. Topol. 12 (2008), 299–350
(arXiv: 0601443v3)
[Juh08] , The sutured Floer homology polytope, Geom. Topol. 14 (2010), 1303–1354
(arXiv: 0802.3415v3)
[Kau83] L. Kauffman, Formal Knot Theory, Princeton University Press (1983)
[Ken12] K.G.Kennedy, A diagrammatic multivariate Alexander invariant of tangles, arXiv: 1205.5781v2
[Kho99] M.Khovanov, A categorification of the Jones polynomial , Duke Math. J. 101 (2000), no. 3,
359–426 (arXiv: 9908171v2)
[KM10] P.Kronheimer, T.Mrowka, Instanton Floer homology and the Alexander polynomial , Algebr.
Geom. Topol. 10 (2010), no. 3, 1715–1738
[KR04] M.Khovanov, L.Rozansky, Matrix factorizations and link homology , arXiv: 0401268v2
[L16] P. Lambert-Cole, Twisting, mutation and knot Floer homology , arXiv: 1608.02011
[L17] , On Conway mutation and link homology , arXiv: 1701.00880
[Lic97] W.B.R. Lickorish, An Introduction to Knot Theory, Springer (1997)
[LM87] W.B.R. Lickorish, K.C.Millett, A polynomial invariant of oriented links, Topol. 26 (1987),
107–141
[Lip05] R. Lipshitz, A cylindrical reformulation of Heegaard Floer homology , Geom. Topol. 10 (2006),
955–1096 (arXiv: 0502404v2)
[LOT08] R. Lipshitz, P.Ozsváth, D.Thurston, Bordered Heegaard Floer homology: Invariance and
pairing , arXiv: 0810.0687v5
[LOT10] , Heegaard Floer homology as morphism spaces, Quantum Topol. 2 (2011), no. 4,
381–449, (arXiv: 1005.1248v2)
[Man06] C.Manolescu, An unoriented skein exact triangle for knot Floer homology , Math. Res. Lett. 14
(2007), 839–852 (arXiv: 0609.531v3)
[OS01] P.Ozsváth, Z. Szabó, Holomorphic discs and topological invariants of closed 3-manifolds, Ann.
Math. 159 (2004), 1027–1158 (arXiv: 0101206v4)
[OS02] , Heegaard Floer homology and alternating knots, Geom. Topol. 7 (2003), 225–254
(arXiv: 0209149v3)
[OS03a] , Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58–116
(arXiv: 0209056v4)
[OS03b] , Knot Floer homology, genus bounds, and mutation, arXiv: 0303225v2
[OS05] , Holomorphic disks and link invariants, Algebr. Geom. Topol. 8 (2008), 615–692
(arXiv: 0512286v2)
[OS06a] , Link Floer homology and the Thurston norm, arXiv: 0601618v3
[OS06b] , Heegaard diagrams and Floer homology , Proc. ICM 2 (2006), no. 51, 1083–1099
(arXiv: 0602232v1)
[OS07] , A cube of resolutions for knot Floer homology , J. Topol. 2 (2009), no. 4, 865–910
(arXiv: 0705.3852v1)
[OS16] , Kauffman states, bordered algebras, and a bigraded knot invariant , arXiv: 1603.06559v1
[OSS07] P.Ozsváth, A. I. Stipsicz, Z. Szabó, Floer homology and singular knots, J. Topol. 2 (2009),
380–404 (arXiv: 0705.2661v3)
[Pol10] M.Polyak, Alexander-Conway invariants of tangles, arXiv: 1011.6200v1
[PV14] I. Petkova, V.Vértesi, Combinatorial tangle Floer homology , arXiv: 1410.2161v2
[Ras03] J. A.Rasmussen, Floer homology and knot complements, PhD thesis (2003), Harvard
(arXiv: 0306378v1)
BIBLIOGRAPHY 123
[Ras05] , Knot polynomials and knot homologies, arXiv: 0504045v1
[Rie14] E.Riehl, Categorical homotopy theory , Cambridge University Press (2014)
[Srk06] S. Sarkar, Maslov index formulas for Whitney n-gons, J. Sympl. Geom. 9 (2011), no. 2, 251–270
(arXiv: 0609673v4)
[Srt13] A. Sartori, The Alexander polynomial as quantum invariant of links, Ark. Mat. 53 (2015),
177–202 (arXiv: 1308.2047v2)
[Sei08] P. Seidel, Fukaya Categories and Picard-Lefschetz Theory, Zürich Lectures in Advanced
Mathematics, European Math. Soc. (2008)
[Weh09] S.M.Wehrli, Mutation invariance of Khovanov homology over F2, arXiv: 0904.3401v1
[Zar09] R. Zarev, Bordered Floer homology for sutured manifolds, arXiv: 0908.1106v2
[Zem16] I. Zemke, Link cobordisms and functoriality in link Floer homology , arXiv: 1610.05207v1
[Zib16] C.B. Zibrowius, On a polynomial Alexander invariant for tangles and its categorification,
arXiv: 1601.04915v1
[nLab2] entry for curved dg-algebra in nLab: https://ncatlab.org/nlab/show/curved+dg-algebra
[nLab1] entry for enriched categories in nLab: https://ncatlab.org/nlab/show/enriched+category
This thesis is available on arXiv, along with the files below. (arXiv: 1610.07494)
[APT.m] Mathematica package APT.m for computing ∇sT
[BSFH.m] Mathematica package BSFH.m for computing bordered sutured Floer homology
[APT.nb] Mathematica notebook APT.nb
[nb1] Mathematica notebook 2m3ptBSD.nb
[nb2] Mathematica notebook 2m3ptmutBSD.nb
[nb3] Mathematica notebook ClosingBSAx2.nb
[nb4] Mathematica notebook GlueingBSAAx4e.nb
Furthermore, scans of hand-drawn Heegaard diagrams used for the computations in the
Mathematica notebooks [nb1], [nb2] and [nb4] can be found under the address
http://www.dpmms.cam.ac.uk/~cbz20/documents/research/HDs.pdf.