On some $p$-differential graded link homologies II

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.