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

In this paper, we introduce two monoidal supercategories: the odd dotted Temperley–Lieb category [Formula: see text], which is a generalization of the odd Temperley–Lieb category studied by Brundan and Ellis in [Monoidal Supercategories, Commun. Math. Phys. 351 (2017) 1045–1089], and the odd annular Bar-Natan category [Formula: see text], which generalizes the odd Bar-Natan category studied by Putyra in [A [Formula: see text]-category of chronological cobordisms and odd Khovanov homology, Banach Center Publ. 103 (2014) 291–355]. We then show there is an equivalence of categories between them if [Formula: see text]. We use this equivalence to better understand the action of the Lie superalgebra [Formula: see text] on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and the second author in [An action of [Formula: see text] on odd annular Khovanov homology, Math. Res. Lett. 27(3) (2020) 711–742].