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Quantum link homology via trace functor I


Beliakova, Anna; Putyra, Krzysztof K; Wehrli, Stephan M (2019). Quantum link homology via trace functor I. Inventiones Mathematicae, 215(2):383-492.

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 $q$.

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 $q$.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:General Mathematics
Language:English
Date:1 February 2019
Deposited On:17 Jan 2019 09:30
Last Modified:09 Feb 2019 02:06
Publisher:Springer
ISSN:0020-9910
Additional Information:This is a post-peer-review, pre-copyedit version of an article published in Inventiones mathematicae. The final authenticated version is available online at: https://doi.org/10.1007%2Fs00222-018-0830-0
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00222-018-0830-0

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