Khovanov homology and categorification of skein modules

TL;DR

This paper constructs a functorial Khovanov homology for links in thickened surfaces, categorifying the gl(2) skein module and exploring its algebraic and monoidal structures, with implications for basis and positivity conjectures.

Contribution

It introduces a categorification of the gl(2) skein module for surface links, including a functorial homology, a candidate monoidal structure, and spectral sequences relating surface link homologies.

Findings

Constructed a functorial Khovanov homology for surface links.

Proposed a categorification of the gl(2) skein module using gl(2) foams.

Provided evidence for the monoidality conjecture and spectral sequences between surface link homologies.

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 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 gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the 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 gl(2) skein algebra and verify an analogue of a positivity conjecture of Fock--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.