论文标题
Khovanov同源性的第一个Vassiliev衍生物的分解及其应用
Decomposition of the first Vassiliev derivative of Khovanov homology and its application
论文作者
论文摘要
Khovanov的同源性通过Vassiliev Skein关系的分类类似物扩展到奇异链接。鉴于Vassiliev理论，扩展的Khovanov同源性可以看作是Khovanov同源性的Vassiliev衍生物。在本文中，我们开发了一种计算第一个衍生物的新方法。也就是说，我们引入了一种称为Crux复合物的复合物，并证明具有独特双点的奇异链接的Khovanov同源物与Crux Complecters上的内态性的Cofibers同型。由于Crux复合物实际上对于某些链接很小，因此结果可以直接计算Khovanov同源性的第一个衍生物。此外，它与分类的Vassiliev Skein关系一起为Khovanov同源性计算提供了一种全新的方法。实际上，我们将结果应用于以通用方式确定所有扭曲结的Khovanov复合物。
Khovanov homology extends to singular links via a categorified analogue of Vassiliev skein relation. In view of Vassiliev theory, the extended Khovanov homology can be seen as Vassiliev derivatives of Khovanov homology. In this paper, we develop a new method to compute the first derivative. Namely, we introduce a complex, called a crux complex, and prove that the Khovanov homologies of singular links with unique double points are homotopic to cofibers of endomorphisms on crux complexes. Since crux complexes are actually small for some links, the result enables a direct computation of the first derivative of Khovanov homology. Furthermore, it together with a categorified Vassiliev skein relation provides a brand-new method for the computation of Khovanov homology. In fact, we apply the result to determine the Khovanov complexes of all twist knots in a universal way.
