A categorification of the colored Jones polynomial at a root of unity

TL;DR

This paper introduces a new categorification of the colored Jones polynomial at roots of unity by leveraging a p-differential structure on triply-graded homology, connecting knot invariants with homotopy theory.

Contribution

It develops a novel p-differential framework on Khovanov--Rozansky homology, leading to a categorification of the colored Jones polynomial at specific roots of unity.

Findings

Establishes a p-differential on homology over fields of positive characteristic.

Shows compatibility of Cautis's differential with the p-differential structure.

Provides a categorification of the colored Jones polynomial at 2p-th roots of unity.

Abstract

There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.