Geometry and Topology Seminar

Khovanov homology is one of the most popular tools used to study links in $S^3$.  If the link is in a thickened annulus, there is an annular refinement of Khovanov homology that contains additional structure.  In particular, 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 $sl_2$.  We will discuss how this structure can be understood in the setting of $L_\infty$-algebras and modules.  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, and the higher $L_\infty$-operations can be computed using explicit formulas.