A new presentation of the osp(1|2)-polynomial link invariant and categorification

TL;DR

This paper introduces a new skein-based presentation of the osp(1|2)-polynomial link invariant, connects it to existing invariants, and demonstrates its categorification via a modified Khovanov homology with a $Z_4$-grading.

Contribution

It provides a novel skein relation for the osp(1|2)-link invariant and constructs its categorification using a modified Khovanov homology, linking it to existing invariants and homological theories.

Findings

The new invariant coincides with Clark's osp(1|2)-link invariant.

The invariant can be categorified using a modified Khovanov homology with a $Z_4$-grading.

A modified version of Putyra's covering Khovanov homology is constructed.

Abstract

There is a known connection between the osp(1|2n) polynomial knot invariant $J_K^n$ and the so(2n+1) knot invariant ${}_{so} J_K^n$ studied by Clark in arXiv:1509.03533 and Blumen in arXiv:0901.3232. In the rank one case, the uncolored $U_{q}(osp(1|2))$ link invariant is equal to the $U_{t^{-1}q}(sl_2)$ link invariant where $t^2=-1$. We define a skein relation similar to the Kauffman bracket, and use that to recover an oriented link invariant which coincides with Clark's uncolored osp(1|2)-link invariant. This definition also comes from the representation theory of $U_{q,\pi}(sl_2)$, but using different methods from Clark. We show that our invariant is easily categorified by a slightly modified version of Khovanov homology equipped with an extra $\mathbb{Z}_4$-grading. We also construct a similarly modified version of Putyra's covering Khovanov homology from arXiv:1310.1895. This suggests that the similarity between the two invariants holds at the categorified level as well.