A local $\mathfrak{gl}_{1|1}$-action on odd Khovanov homology

Abstract

We show that odd Khovanov homology carries an action of the super Lie algebra $\mathfrak{gl}_{1|1}$, given extra choice of markings on the link. Moreover, we show that this action arises from an action on super $\mathfrak{gl}_{2}$-foams, in the extended-TQFT framework developed by the second author and Vaz; in particular, it extends to tangles. Finally, we relate the action to torsion $\mathbb{Z}/n\mathbb{Z}$ in pretzel links $P(n,n,-n)$. In particular, this shows that all torsion can appear in odd Khovanov homology.