Sutured annular Khovanov-Rozansky homology

TL;DR

This paper develops a new sl(n) homology theory for knots and links in the annulus, extending sutured annular Khovanov homology to a more general setting with Lie algebra actions.

Contribution

It introduces sutured annular Khovanov-Rozansky homology, extending previous theories and connecting to Lie algebra actions via trace decategorifications and categorified quantum groups.

Findings

Defines sutured annular Khovanov-Rozansky homology for annular links.

Shows the homology carries an sl(n) action, generalizing previous results.

Recovers known results in the case n=2.

Abstract

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which in the n=2 case recovers a result of Grigsby-Licata-Wehrli.