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.

arXiv comment42 pages, 32 figures

References in corpus (5)

• Categorification of the Kauffman bracket skein module of I-bundles over surfaces
• On knot Floer homology in double branched covers
• Annular Khovanov homology and knotted Schur-Weyl representations
• A rank inequality for the knot Floer homology of double branched covers
• Equivariant Jones Polynomials of periodic links

Cited by in corpus (3)

• Categorical lifting of the Jones polynomial: a survey
• Equivariant Khovanov Homology of Periodic Links
• Localization in Khovanov homology