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 $C$ in new window and endobifunctor $\sum : C \rightarrow C$ . For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $\sum_q$ such that $\sum\nolimits_{q}a:= q−^{deg α} \sum α$ for any 2-morphism α and coincides with $\sum$ 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 homology for links in a thickened annulus. We prove that our homology carries an action of $U_q (sI_2)$ , 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 depend on the quantum parameter $q$.