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Annular Khovanov homology and knotted Schur–Weyl representations

Part of: Low-dimensional topology Groups and algebras in quantum theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  28 November 2017

J. Elisenda Grigsby ,
Anthony M. Licata  and
Stephan M. Wehrli
J. Elisenda Grigsby
Affiliation:
Boston College, Department of Mathematics, 5th floor Maloney, Chestnut Hill, MA 02467, USA email grigsbyj@bc.edu
Anthony M. Licata
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, Australia email anthony.licata@anu.edu.au
Stephan M. Wehrli
Affiliation:
Syracuse University, Department of Mathematics, 215 Carnegie, Syracuse, NY 13244, USA email smwehrli@syr.edu
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Abstract

Let $\mathbb{L}\subset A\times I$ be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of $\mathfrak{sl}_{2}(\wedge )$ , the exterior current algebra of $\mathfrak{sl}_{2}$ . When $\mathbb{L}$ is an $m$ -framed $n$ -cable of a knot $K\subset S^{3}$ , its sutured annular Khovanov homology carries a commuting action of the symmetric group $\mathfrak{S}_{n}$ . One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical $\mathfrak{sl}_{2}$ Schur–Weyl duality when $K$ is the Seifert-framed unknot.

MSC classification

Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 81R50: Quantum groups and related algebraic methods 20F36: Braid groups; Artin groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 3 , March 2018 , pp. 459 - 502
Copyright
© The Authors 2017 

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