Quantum Link Homology via Trace Functor I

Abstract

Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory $\mathbf{C}$ and endobifunctor $\Sigma :\mathbf{C}\to \mathbf{C}$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor ${\Sigma }_q$ such that ${\Sigma }_q\alpha :=q^{-°\alpha }\Sigma \alpha$ for any 2-morphism $\alpha$ and coincides with $\Sigma$ otherwise. Applying the quantized trace to the~bicategory of Chen-Khovanov bimodules we get a new triply graded link homology theory called quantum annular link homology. If $q=1$ we reproduce Asaeda-Przytycki-Sikora (APS) homology for links in a thickened annulus. We prove that our homology carries an action of ${\mathcal{U}}_q({\mathfrak{sl}}_2)$, which intertwines the action of cobordisms. In particular, the~quantum annular homology of an $n$-cable admits an action of the braid group, which commutes with the quantum group action and factors through the Jones skein relation. This produces a nontrivial invariant for surfaces knotted in four dimensions. Moreover, a direct computation for torus links shows that the rank of quantum annular homology groups does depend on the quantum parameter $q$.

A major revision of the previous version (functoriality of traces and shadows explained, construction of traces and shadows on (bi)categories of complexes, etc.); 85 pages, color figures (but can be safely printed black and white)