Annular Khovanov homology and knotted Schur–Weyl representations. (28th November 2017)

• Record Type: Journal Article
• Authors: Grigsby, J. Elisenda; Licata, Anthony M.; Wehrli, Stephan M.
• Is Part Of: Compositio mathematica. Volume 154:Number 3(2018)
• Page Start: 459
• Page End: 502
• Publication Date: 2017-11-28
• Subjects: 57M27 (primary), -- 81R50, -- 20F36 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X17007540
• Languages: English
• ISSNs: 0010-437X

Abstract

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.