On some $p$-differential graded link homologies II

TL;DR

This paper introduces a family of link homologies that generalize previous categorifications of the Jones polynomial at roots of unity, revealing non-isomorphic invariants for specific parameter choices.

Contribution

It extends the categorification framework by specializing the Cautis differential to a broader class of integers, producing new link homologies with distinct invariants.

Findings

For even k, the homologies categorify the Jones polynomial at 2p-th roots of unity.

The new homologies are non-isomorphic invariants despite categorifying the same polynomial.

Generalization of previous categorification methods to a wider family of differentials.

Abstract

In arXiv:2009.06498, a link invariant categorifying the Jones polynomial at a $2p$th root of unity, where $p$ is an odd prime, was constructed. This categorification utilized an $N=2$ specialization of a differential introduced by Cautis. Here we give a family of link homologies where the Cautis differential is specialized to a positive integer of the form $N=kp+2$. When $k$ is even, all these link homologies categorify the Jones polynomial evaluated at a $2p$th root of unity, but they are non-isomorphic invariants.