An $L_\infty$-module Structure on Annular Khovanov Homology

TL;DR

This paper enhances the algebraic structure of annular Khovanov homology by establishing an $L_$-module framework over the exterior current algebra of $sl_2$, including explicit formulas for higher operations.

Contribution

It introduces an $L_$-algebra and module structure on annular Khovanov homology, extending previous $sl_2()$ actions to a homotopy algebra setting with invariance properties.

Findings

$L_$-structure on annular Khovanov homology established

Explicit formulas for higher $L_$-operations provided

Invariance under Reidemeister moves up to $L_$-quasi-isomorphism

Abstract

Let $L$ be a link in a thickened annulus. Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $sl_2(\wedge)$, the exterior current algebra of the Lie algebra $sl_2$. In this paper, we upgrade this result to the setting of $L_\infty$-algebras and modules. That is, we show that $sl_2(\wedge)$ is an $L_\infty$-algebra and that the annular Khovanov homology of $L$ is an $L_\infty$-module over $sl_2(\wedge)$. Up to $L_\infty$-quasi-isomorphism, this structure is invariant under Reidemeister moves. Finally, we include explicit formulas to compute the higher $L_\infty$-operations.