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

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 $s{l}_2(\wedge )$, the exterior current algebra of the Lie algebra $s{l}_2$. In this paper, we upgrade this result to the setting of $L_{\mathrm{\infty }}$-algebras and modules. That is, we show that $s{l}_2(\wedge )$ is an $L_{\mathrm{\infty }}$-algebra and that the annular Khovanov homology of $L$ is an $L_{\mathrm{\infty }}$-module over $s{l}_2(\wedge )$. Up to $L_{\mathrm{\infty }}$-quasi-isomorphism, this structure is invariant under Reidemeister moves. Finally, we include explicit formulas to compute the higher $L_{\mathrm{\infty }}$-operations.