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# Mathematics > Geometric Topology

# Title:Quantum Link Homology via Trace Functor I

(Submitted on 11 May 2016 (v1), last revised 27 Sep 2018 (this version, v2))

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 $Σ\colon \mathbf C \to\mathbf C$. For a graded linear bicategory and a fixed invertible parameter $q$, we quantize this theory by using the endofunctor $Σ_q$ such that $Σ_q α:=q^{-° α}Σα$ for any 2-morphism $α$ and coincides with $Σ$ 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 $q$.

## Submission history

From: Krzysztof Putyra [view email]**[v1]**Wed, 11 May 2016 17:24:36 UTC (125 KB)

**[v2]**Thu, 27 Sep 2018 16:12:35 UTC (150 KB)