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

TL;DR

This paper establishes a spectral sequence relating the annular Khovanov homology of 2-periodic links and their quotients, leading to a rank inequality and exploring implications for Khovanov homology.

Contribution

It introduces a spectral sequence that connects the annular Khovanov homologies of 2-periodic links and their quotients, proving a new rank inequality.

Findings

Spectral sequence linking AKh of 2-periodic links and quotients.

Proved rank inequality for AKh in quantum and $sl_2$ gradings.

Discussed decategorified consequences and partial results for Khovanov homology.

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 $sl_2$ weight space gradings, proving a rank inequality $rank\ AKh^{j,k}(L) \leq rank\ AKh^{2j-k,k} (\tilde L)$ for every pair of quantum and $sl_2$ 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.