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Quantum Topology

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Volume 12, Issue 1, 2021, pp. 129–209
DOI: 10.4171/QT/148

Published online: 2021-03-17

Khovanov homology and categorification of skein modules

Hoel Queffelec[1] and Paul Wedrich[2]

(1) Université de Montpellier, France
(2) Max-Planck-Institut für Mathematik, Bonn, Germany and Australian National University, Canberra, Australia

For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding $\mathfrak {gl}_2$ skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of $\mathfrak {gl}_2$ foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the $\mathfrak {gl}_2$ skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunctor on the target category and conjecture that it extends to a monoidal structure. This would give rise to a canonical basis of the associated $\mathfrak {gl}_2$ skein algebra and verify an analogue of a positivity conjecture of Fock and Goncharov and Thurston. We provide evidence towards the monoidality conjecture by checking several instances of a categorified Frohman–Gelca formula for the skein algebra of the torus. Finally, we recover a variant of the Asaeda–Przytycki–Sikora surface link homologies and prove that surface embeddings give rise to spectral sequences between them.

Keywords: Skein module, Khovanov homology, knots in thickened surfaces

Queffelec Hoel, Wedrich Paul: Khovanov homology and categorification of skein modules. Quantum Topol. 12 (2021), 129-209. doi: 10.4171/QT/148