2022/10/18 by Mark Ebert, Ebert, Mark
Mathematics · Medicine · #17A70 #17B37 #20G42 #57K14 #57K18 #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2210.09583
openalex publication_date 2022/10/18 · openalex created_date 2022/10/22 · openalex updated_date 2026/07/28
There is a known connection between the osp(1|2n) polynomial knot invariant JKn and the so(2n+1) knot invariant so JKn studied by Clark in arXiv:1509.03533 and Blumen in arXiv:0901.3232. In the rank one case, the uncolored Uq(osp(1|2)) link invariant is equal to the Ut-1q(sl2) link invariant where t2=-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 Uq,π(sl2), 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 ℤ4-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.