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 $E^1\cong AKh(\tilde{L}){\otimes }_{{\mathbb{F}}_2}{\mathbb{F}}_2[\theta ,{\theta }^{-1}]\rightrightarrows E^{\infty }\cong AKh(L){\otimes }_{{\mathbb{F}}_2}{\mathbb{F}}_2[\theta ,{\theta }^{-1}].$ This spectral sequence splits along quantum and $s{l}_2$ weight space gradings, proving a rank inequality $rankAK{h}^{j,k}(L)\le rankAK{h}^{2j-k,k}(\tilde{L})$ for every pair of quantum and $s{l}_2$ weight space gradings $(j,k)$. We also present a few decategorified consequences and discuss partial results toward a similar statement for the Khovanov homology of 2-periodic links.

42 pages, 32 figures

Cited by in corpus (2)

• Categorical lifting of the Jones polynomial: a survey
• Localization in Khovanov homology