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

Abstract

Asaeda, Przytycki, and Sikora (2004) 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 $\{0\}\times T^2$. We also prove that Asaeda–Przytycki–Sikora homology detects the unlink and torus links in $(-1,1)\times T^2$.