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

Abstract

We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category ${\mathcal{T}\mathcal{L}}_{o,\bullet }(\delta )$, which is a generalization of the odd Temperley-Lieb category studied by Brundan and Ellis, and the odd annular Bar-Natan category ${\mathcal{B}\mathcal{N}}_o(\mathbb{A})$, 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.

83 pages