Odd annular Bar-Natan category and gl(1|1)

TL;DR

This paper introduces two related monoidal supercategories generalizing odd Temperley-Lieb and Bar-Natan categories, establishing their equivalence at zero parameter, and applies this to better understand the action of rak{gl}(1|1) on odd Khovanov homology.

Contribution

It constructs and relates new supercategories that extend existing categories and applies this framework to analyze Lie superalgebra actions on knot homologies.

Findings

Established an equivalence between the odd dotted Temperley-Lieb and odd annular Bar-Natan categories at elta=0.

Provided a new perspective on the rak{gl}(1|1) action on odd Khovanov homology.

Extended the understanding of supercategory structures in knot theory.

Abstract

We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category $\mathcal{T\!L}_{o,\bullet}(\delta)$, which is a generalization of the odd Temperley-Lieb category studied by Brundan and Ellis, and the odd annular Bar-Natan category $\mathcal{BN}_{\!o}(\mathbb{A})$, which generalizes the odd Bar-Natan category studied by Putyra. We then show there is an equivalence of categories between them if $\delta=0$. We use this equivalence to better understand the action of the Lie superalgebra $\mathfrak{gl}(1|1)$ on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and the second author.