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

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.