Quantum Link Homology via Trace Functor I

TL;DR

This paper develops a new quantum link homology theory using trace functors in bicategories, extending classical link invariants to a quantum setting with applications to knotted surfaces.

Contribution

It introduces quantum annular link homology via trace functors, generalizing APS homology and incorporating quantum group actions for the first time.

Findings

Quantum annular homology extends APS homology at q=1.

Homology groups depend on the quantum parameter q.

Braid group actions commute with quantum group actions.

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 $\Sigma\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $\Sigma_q$ such that $\Sigma_q \alpha:=q^{-\deg \alpha}\Sigma\alpha$ 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 $q$.