$\mathfrak{sl}_2$缠结同调上实现交叉变换的配边法和范畴化的Vassiliev绞关系
A cobordism realizing crossing change on $\mathfrak{sl}_2$ tangle homology and a categorified Vassiliev skein relation
摘要：
本文利用配边的方法讨论了Khovanov同调的0度交叉变换。也就是说，使用Bar-Natan的Khovanov同调的形式主义，我们引入了一个协边的总和，它在两个交叉变化图的复形上产生一个态射，我们称之为“亏格一态射”。“证明了态射在缠结图中的二重点的移动下是不变性的。因此，在Vassiliev理论的精神下，采用迭代映射锥，我们得到了扩展sl（2）tangle同调的奇异tangle的不变量;例子包括Lee同调，Bar-Natan同调，Naot的泛Khovanov同调以及任意系数的Khovanov同调。我们还验证了不变量满足范畴化的类似物的Vassiliev绞关系和FI关系。
英文摘要：
In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Natan homology, and Naot's universal Khovanov homology as well as Khovanov homology with arbitrary coefficients. We also verify that the invariant satisfies categorified analogues of Vassiliev skein relation and the FI relation.
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