Instantons and Khovanov skein homology on $I\times T^2$

TL;DR

This paper characterizes when the Asaeda-Przytycki-Sikora homology of links in a product of an interval and a torus has rank 2, showing it occurs precisely for knots isotopic to those in a specific torus slice.

Contribution

It establishes a precise topological criterion linking the homology rank to isotopy classes of knots in a torus product, extending understanding of Khovanov-type invariants.

Findings

Homology rank 2 iff the link is isotopic to a knot in a specific torus slice

Provides a characterization of links with minimal homology in this setting

Connects homological properties to isotopy classes in a 3-manifold

Abstract

Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in $I$-bundles over compact surfaces. We prove that for a link $L\subset (-1,1)\times T^2$, the Asaeda-Przytycki-Sikora homology of $L$ has rank $2$ with $\mathbb{Z}/2$-coefficients if and only if $L$ is isotopic to an embedded knot in $\{0\}\times T^2$.