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

Zhang, Melissa

doi:10.2140/agt.2018.18.1147

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A rank inequality for the annular Khovanov homology of $2\mskip-1.5mu$–periodic links

Melissa Zhang

Algebraic & Geometric Topology 18 (2018) 1147–1194

DOI: 10.2140/agt.2018.18.1147

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} ≅ AKh (\tilde{L}) \otimes_{F} F [θ, θ^{- 1}] ⇉ E^{\infty} ≅ AKh (L) \otimes_{F} F [θ, θ^{- 1}] .

This spectral sequence splits along quantum and ${s l}_{2}$ weight-space gradings, proving a rank inequality $rk {AKh}^{j, k} (L) \leq rk {AKh}^{2 j - k, k} (\tilde{L})$ 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, as well as some frameworks for obstructing $2$ –periodicity in links.

Keywords

Khovanov homology, periodic links, localization

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Secondary: 57M60

References

Publication

Received: 14 July 2017

Revised: 2 November 2017

Accepted: 25 November 2017

Published: 12 March 2018

Authors

Melissa Zhang
Department of Mathematics Boston College Chestnut Hill, MA United States

Volume 18, issue 2 (2018)