Instanton homology and knot detection on thickened surfaces

TL;DR

This paper proves that the APS homology detects the unknot in certain thickened surfaces, providing a new detection result for generalized Khovanov homology on an infinite family of manifolds.

Contribution

It establishes the first detection result for generalized Khovanov homology on an infinite family of manifolds, using instanton homology techniques different from previous methods.

Findings

APS homology of L has rank 2 iff L is isotopic to a knot in a specific slice

APS homology detects the unknot in the given manifold

Characterizes links with minimal sutured instanton homology

Abstract

Suppose $\Sigma$ is a compact oriented surface (possibly with boundary) that has genus zero, and L is a link in the interior of $(-1,1)\times\Sigma$. We prove that the Asaeda-Przytycki-Sikora (APS) homology of L has rank 2 if and only if L is isotopic to an embedded knot in $\{0\}\times\Sigma$. As a consequence, the APS homology detects the unknot in $(-1,1)\times\Sigma$. This is the first detection result for generalized Khovanov homology that is valid on an infinite family of manifolds, and it partially solves a conjecture in arxiv:2005.12863. Our proof is different from the previous detection results obtained by instanton homology because in this case, the second page of Kronheimer-Mrowka's spectral sequence is not isomorphic to the APS homology. We also characterize all links in product manifolds that have minimal sutured instanton homology, which may be of independent interest.