We introduce two monoidal supercategories: the odd dotted Temperley-Lieb category TLo,•(δ), which is a generalization of the odd Temperley-Lieb category studied by Brun-dan and Ellis in [5], and the odd annular Bar-Natan category BNo(A), which generalizes the odd Bar-Natan category studied by Putyra in [16]. We then show there is an equivalence of categories between them if δ=0. We use this equivalence to better understand the action of the Lie superalgebra gl(1|1) on the odd Khovanov homology of a knot in a thickened annulus found by Grigsby and Wehrli in [7]
