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

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 $U_q(osp(1\mathrm{\mid }2))$ link invariant is equal to the $U_{t^{-1}q}(s{l}_2)$ 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 $U_{q,\pi }(s{l}_2)$, 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.