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

Abstract

Kronheimer-Mrowka conjectured that sutured instanton Floer homology $SHI(M,\gamma)$ has the same dimension as the sutured Floer homology $SFH(M,\gamma)$ constructed by Juh\'{a}sz for any balanced sutured manifold $(M,\gamma)$. 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 $I^\sharp(Y)$. 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 $S^3$.

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 spin$^c$ 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 $KHI(S^3,K)$ to $I^\sharp(S^3_n(K))$ by a large surgery formula, where $S^3_n(K)$ is obtained from a knot $K\subset S^3$ by $n$-Dehn surgery. As an application, we show that $S^3_r(K)$ 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 $\Delta_K(t)$ lie in $\{-1,0,1\}$. In particular, any hyperbolic alternating knot satisfies this property.