An enhanced Euler characteristic of sutured instanton homology

TL;DR

This paper introduces an enhanced Euler characteristic for sutured instanton homology, providing new decomposition techniques, better bounds on dimensions, and applications to knot detection and conjectures in low-dimensional topology.

Contribution

It develops a decomposition of sutured instanton homology based on torsions, defines an enhanced Euler characteristic, and applies these to knot detection and conjecture verification.

Findings

Enhanced Euler characteristic equals the standard Euler characteristic for sutured instanton homology.

Provides a lower bound on the dimension of sutured instanton homology.

Proves instanton knot homology detects the unknot in L-spaces and verifies a conjecture for certain knots.

Abstract

For a balanced sutured manifold $(M,\gamma)$, we construct a decomposition of $SHI(M,\gamma)$ with respect to torsions in $H=H_1(M;\mathbb{Z})$, which generalizes the decomposition of $I^\sharp(Y)$ in previous work of the authors. This decomposition can be regarded as a candidate for the counterpart of the torsion spin$^c$ decompositions in $SFH(M,\gamma)$. Based on this decomposition, we define an enhanced Euler characteristic $\chi_{\rm en}(SHI(M,\gamma))\in\mathbb{Z}[H]/\pm H$ and prove that $\chi_{\rm en}(SHI(M,\gamma))=\chi(SFH(M,\gamma))$. This provides a better lower bound on $\dim_\mathbb{C}SHI(M,\gamma)$ than the graded Euler characteristic $\chi_{\rm gr}(SHI(M,\gamma))$. As applications, we prove instanton knot homology detects the unknot in any instanton L-space and show that the conjecture $KHI(Y,K)\cong \widehat{HFK}(Y,K)$ holds for all $(1,1)$-L-space knots and constrained knots in lens spaces, which include all torus knots and many hyperbolic knots in lens spaces.