Annular Khovanov homology and knotted Schur-Weyl representations

TL;DR

This paper demonstrates that sutured annular Khovanov homology of links in a thickened annulus admits actions of the exterior current algebra of sl_2 and the symmetric group, leading to a knotted Schur-Weyl duality.

Contribution

It introduces a new algebraic structure on sutured annular Khovanov homology, connecting it with current algebra actions and symmetric group representations.

Findings

Sutured annular Khovanov homology admits an exterior current algebra of sl_2 action.

For cable links, it carries a commuting symmetric group action.

The resulting knotted Schur-Weyl representation generalizes classical duality.

Abstract

Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a "knotted" Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.