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

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