2020/07/31 by Jun Yoshida, Yoshida, Jun · 1 citation
Mathematics · Medicine · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Botulinum Toxin and Related Neurological Disorders
paper · pdf · doi:10.48550/arxiv.2007.15867
Khovanov homology extends to singular links via a categorified analogue of\nVassiliev skein relation. In view of Vassiliev theory, the extended Khovanov\nhomology can be seen as Vassiliev derivatives of Khovanov homology. In this\npaper, we develop a new method to compute the first derivative. Namely, we\nintroduce a complex, called a crux complex, and prove that the Khovanov\nhomologies of singular links with unique double points are homotopic to\ncofibers of endomorphisms on crux complexes. Since crux complexes are actually\nsmall for some links, the result enables a direct computation of the first\nderivative of Khovanov homology. Furthermore, it together with a categorified\nVassiliev skein relation provides a brand-new method for the computation of\nKhovanov homology. In fact, we apply the result to determine the Khovanov\ncomplexes of all twist knots in a universal way.\n