Algebraic & Geometric Topology

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

Melissa Zhang

Abstract

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

$E 1 ≅ AKh ( L ̃ ) ⊗ F F [ θ , θ − 1 ] ⇉ E ∞ ≅ AKh ( L ) ⊗ F F [ θ , θ − 1 ] .$

This spectral sequence splits along quantum and $s l 2$ weight-space gradings, proving a rank inequality $rk AKh j , k ( L ) ≤ rk AKh 2 j − k , k ( L ̃ )$ 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.

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 2 (2018), 1147-1194.

Dates
Revised: 2 November 2017
Accepted: 25 November 2017
First available in Project Euclid: 22 March 2018

https://projecteuclid.org/euclid.agt/1521684033

Digital Object Identifier
doi:10.2140/agt.2018.18.1147

Mathematical Reviews number (MathSciNet)
MR3773751

Zentralblatt MATH identifier
06859617

Citation

Zhang, Melissa. A rank inequality for the annular Khovanov homology of $2$–periodic links. Algebr. Geom. Topol. 18 (2018), no. 2, 1147--1194. doi:10.2140/agt.2018.18.1147. https://projecteuclid.org/euclid.agt/1521684033

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