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

TL;DR

The authors construct a ${\mathfrak{gl}_{1|1}}$-action on odd Khovanov homology by implementing an action of ${\mathfrak{gl}_{1|1}}$ on the super ${\mathfrak{gl}_{2}}$-foams that underlie odd Khovanov homology. This action depends on a choice of markings on tangles and extends to tangles through the extended-TQFT framework, while remaining compatible with the foam-based presentation. They relate the local foam action to the original odd Khovanov construction, showing ${\mathfrak{gl}_{1|1}}$-equivariant correspondence between the foam-based and the classic formulations. A key application is the demonstration that pretzel links of the form $P(nn-n)$ carry a ${\mathfrak{gl}_{1|1}}$-orbit of torsion copies $\mathbb{Z}/n\mathbb{Z}$ in odd Khovanov homology, implying that all torsion arises in the odd theory. The work further clarifies the roles of type X/type Y and situates the construction within graded/covering ${\mathfrak{gl}_{2}}$-Khovanov frameworks, connecting to prior results on torsion phenomena and marking actions in related literature.

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.