Khovanov homology and categorification of skein modules
Hoel QueffelecUniversité de Montpellier, France
Paul WedrichMax-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 skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the 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 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.
Cite this article
Hoel Queffelec, Paul Wedrich, Khovanov homology and categorification of skein modules. Quantum Topol. 12 (2021), no. 1, pp. 129–209DOI 10.4171/QT/148