一种新的osp（1| 2）-多项式连接不变量与范畴化
A new presentation of the osp(1|2)-polynomial link invariant and categorification
摘要：
在osp（1）之间存在已知的连接|2n）多项式结不变量$J_K^n$和Clark在arXiv：1509.03533中研究的so（2n+1）结不变量${}_{so} J_K^n$和Blumen在arXiv：0901.3232中研究的so（2n+1）结不变量${}_{so} J_K^n$。在秩1的情况下，未着色的$U_{q}(osp(1|2))$链接不变量等于$U_{t^{-1}q}(sl_2)$链接不变量，其中$t^2=-1$。我们定义了一个类似于Kauffman括号的skein关系，并使用它来恢复一个与Clark的无色osp（1）相一致的定向链接不变量|2）-链接不变量。这个定义也来自于$U_{q,π}(sl_2)$的表示论，但是使用了与Clark不同的方法。我们表明，我们的不变很容易归类的Khovanov同源性略有修改的版本配备了一个额外的$\mathbb{Z}_4$分级。我们还从arXiv：1310.1895构建了一个类似的修改版本的Putyra覆盖Khovanov同源性。这表明两个不变量之间的相似性在范畴化水平上也成立。
英文摘要：
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 $U_{q,π}(sl_2)$, 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.
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