Khovanov homology and categorification of skein modules

Abstract

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.