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On the functoriality of $\mathfrak{sl}_{2}$ tangle homology

Anna Beliakova, Matthew Hogancamp, Krzysztof K Putyra and Stephan M Wehrli

Algebraic & Geometric Topology 23 (2023) 1303ÔÇô1361

We construct an explicit equivalence between the (bi)category of 𝔤𝔩2 webs and foams and the Bar-Natan (bi)category of Temperley–Lieb diagrams and cobordisms. With this equivalence we can fix functoriality of every link homology theory that factors through the Bar-Natan category. To achieve this, we define web versions of arc algebras and their quasihereditary covers, which provide strictly functorial tangle homologies. Furthermore, we construct explicit isomorphisms between these algebras and the original ones based on Temperley–Lieb cup diagrams. The immediate application is a strictly functorial version of the Beliakova–Putyra–Wehrli quantization of the annular link homology.

Khovanov homology, tangle homology, web, foam
Mathematical Subject Classification
Primary: 57K18
Secondary: 18N25
Received: 15 February 2021
Revised: 4 August 2021
Accepted: 14 September 2021
Published: 6 June 2023
Anna Beliakova
Institut f├╝r Mathematik
Universit├Ąt Z├╝rich
Matthew Hogancamp
Department of Mathematics
Northeastern University
Boston, MA
United States
Krzysztof K Putyra
Institute for Theoretical Studies
ETH Z├╝rich
Stephan M Wehrli
Mathematics Department
Syracuse University
Syracuse, NY
United States

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