Instanton Floer homology, sutures, and Euler characteristics

TL;DR

This paper axiomatizes instanton Floer homology for balanced sutured manifolds, proving its Euler characteristic matches that of sutured Floer homology and relates to classical knot invariants.

Contribution

It provides an axiomatic framework for instanton Floer homology, establishing the equality of Euler characteristics with sutured Floer homology and connecting to the Alexander polynomial.

Findings

Euler characteristic of instanton Floer homology matches that of sutured Floer homology.

The Euler characteristic recovers the multi-variable Alexander polynomial for links.

Examples of knots in lens spaces with matching KHI and HFK dimensions.

Abstract

This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $\chi_{\rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $\chi_{\rm gr}(SHI(M,\gamma))=\chi_{\rm gr}(SFH(M,\gamma))$ for any balanced sutured manifold $(M,\gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $\chi_{\rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $\widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $\mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^\sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $\underline{\rm KHI}^-(Y,K)$ introduced by the first author.