An $L_{\infty }$-module structure on annular Khovanov homology

Abstract

Let $L$ be a link in a thickened annulus. In Grigsby et al. (2018), Grigsby–Licata–Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of ${\mathfrak{sl}}_2(\wedge )$, the exterior current algebra of the Lie algebra ${\mathfrak{sl}}_2$. In this paper, we upgrade this result to the setting of $L_{\infty }$-algebras and modules. That is, we show that ${\mathfrak{sl}}_2(\wedge )$ is an $L_{\infty }$-algebra and that the annular Khovanov homology of $L$ is an $L_{\infty }$-module over ${\mathfrak{sl}}_2(\wedge )$. 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.