On some $p$-differential graded link homologies

TL;DR

This paper demonstrates that triply graded Khovanov-Rozansky homology over fields of positive odd characteristic can be viewed as an invariant in the homotopy category of finite-dimensional p-complexes, leading to a categorification of the Jones polynomial at odd prime roots of unity.

Contribution

It introduces a p-DG structure compatible with the extended differential on triply graded homology, enabling new categorification results for knot invariants.

Findings

Homology descends to an invariant in the homotopy category of p-complexes.

Compatibility of Cautis's p-extended differential with p-DG structure.

Categorification of the Jones polynomial at odd prime roots of unity.

Abstract

We show that the triply graded Khovanov-Rozansky homology of knots and links over a field of positive odd characteristic $p$ descends to an invariant in the homotopy category finite-dimensional $p$-complexes. A $p$-extended differential on the triply graded homology discovered by Cautis is compatible with the $p$-DG structure. As a consequence we get a categorification of the Jones polynomial evaluated at an odd prime root of unity