A rank inequality for the annular Khovanov homology of $2$–periodic links

Abstract

For a $2$–periodic link $\tilde{L}$ in the thickened annulus and its quotient link $L$, we exhibit a spectral sequence with

$$E^1\cong AKh\left(\tilde{L}\right){\otimes }_{\mathbb{F}}\mathbb{F}\left[\theta ,{\theta }^{-1}\right]\rightrightarrows E^{\infty }\cong AKh\left(L\right){\otimes }_{\mathbb{F}}\mathbb{F}\left[\theta ,{\theta }^{-1}\right].$$

This spectral sequence splits along quantum and ${\mathfrak{s}\mathfrak{l}}_2$ weight-space gradings, proving a rank inequality $rk{AKh}^{j,k}\left(L\right)\le rk{AKh}^{2j-k,k}\left(\tilde{L}\right)$ for every pair of quantum and ${\mathfrak{s}\mathfrak{l}}_2$ weight-space gradings $\left(j,k\right)$. We also present a few decategorified consequences and discuss partial results toward a similar statement for the Khovanov homology of $2$–periodic links, as well as some frameworks for obstructing $2$–periodicity in links.