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Mathematics > Geometric Topology

Title: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 $Σ\colon \mathbf C \to\mathbf C$. 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 $q$.
Comments: 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)
Subjects: Geometric Topology (math.GT); Category Theory (math.CT); Quantum Algebra (math.QA)
MSC classes: 57M27, 55N35, 16E40, 18D05, 18F30
Cite as: arXiv:1605.03523 [math.GT]
  (or arXiv:1605.03523v2 [math.GT] for this version)

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)