TY - JOUR
T1 - Odd annular Bar-Natan category and gl(1 | 1)
AU - Necheles, Casey
AU - Wehrli, Stephan
N1 - Publisher Copyright:
© 2024 World Scientific Publishing Company.
PY - 2024/7/1
Y1 - 2024/7/1
N2 - In this paper, we introduce two monoidal supercategories: the odd dotted Temperley-Lieb category TLo,•(σ), 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 BNo(A), which generalizes the odd Bar-Natan category studied by Putyra in [A 2-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 σ = 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 the second author in [An action of gl(1|1) on odd annular Khovanov homology, Math. Res. Lett. 27(3) (2020) 711-742].
AB - In this paper, we introduce two monoidal supercategories: the odd dotted Temperley-Lieb category TLo,•(σ), 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 BNo(A), which generalizes the odd Bar-Natan category studied by Putyra in [A 2-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 σ = 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 the second author in [An action of gl(1|1) on odd annular Khovanov homology, Math. Res. Lett. 27(3) (2020) 711-742].
KW - Annular Khovanov homology
KW - Lie superalgebra
KW - odd Bar-Natan category
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U2 - 10.1142/S0218216524500251
DO - 10.1142/S0218216524500251
M3 - Article
AN - SCOPUS:85204675072
SN - 0218-2165
VL - 33
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
IS - 8
M1 - 2450025
ER -