New techniques in calculation of sutured instanton Floer homology: by Heegaard diagrams, Euler characteristics, and Dehn surgery formulae

Change log
Ye, Fan 

Kronheimer-Mrowka conjectured that sutured instanton Floer homology SHI(M,γ) has the same dimension as the sutured Floer homology SFH(M,γ) constructed by Juh'{a}sz for any balanced sutured manifold (M,γ). Motivated by their conjecture, we introduce new techniques for calculations of sutured instanton Floer homology, some of which are inspired by analogous results in Heegaard Floer theory.

The first technique is based on Heegaard diagrams of balanced sutured manifolds, from which we obtain an upper bound on the dimension of SHI. For any rationally null-homologous knot K in a closed 3-manifold Y, we prove the dimension of the instanton knot homology KHI(Y,K) is greater than or equal to the dimension of the framed instanton homology I(Y). We also use this technique to compute the instanton knot homology of (1,1)-knots that are also L-space knots. In particular, we calculate the homologies for all torus knots in S3.

The second technique is based on the identification of Euler characteristics of SFH and SHI, from which we obtain a lower bound on the dimension of SHI. We construct a decomposition of SHI analogous to the spinc structure decomposition of SFH, and prove that the enhanced Euler characteristic defined by this decomposition equals to the Euler characteristic of SFH. We introduce a family of (1,1)-knots called \textbf{constrained knots} and show that the upper bound from the first technique coincides with the lower bound from the second technique.

The third technique relates KHI(S3,K) to I(Sn3(K)) by a large surgery formula, where Sn3(K) is obtained from a knot KS3 by n-Dehn surgery. As an application, we show that Sr3(K) admits an irreducible SU(2) representation for a dense set of slopes r unless K is a prime knot and the coefficients of the Alexander polynomial ΔK(t) lie in {−1,0,1}. In particular, any hyperbolic alternating knot satisfies this property.

Rasmussen, Jacob
sutured manifold, instanton Floer homology, Heegaard Floer homology
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge