Instanton Floer homology, sutures, and Euler characteristics

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 ${\chi }_{\text{gr}}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that ${\chi }_{\text{gr}}(\mathrm{SHI}(M,\gamma ))={\chi }_{\text{gr}}(\mathrm{SFH}(M,\gamma ))$ for any balanced sutured manifold $(M,\gamma )$. In particular, for any link $L$ in $S^3$, the Euler characteristic ${\chi }_{\text{gr}}(\mathrm{KHI}(S^3,L))$ 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 $\mathrm{KHI}$ and $\widehat{\mathrm{HFK}}$ 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 $\mathrm{KHI}(Y,K)$, the decomposition of $I^{♯}(Y)$ discussed in the previous paper, and the minus version of instanton knot homology ${\underline{\mathrm{KHI}}}^{-}(Y,K)$ introduced by Zhenkun Li.