Decomposition of the first Vassiliev derivative of Khovanov homology and its application

TL;DR

This paper introduces a novel method using crux complexes to compute the first Vassiliev derivative of Khovanov homology, enabling direct calculations and new insights into singular links and twist knots.

Contribution

It develops a new approach with crux complexes to compute the first Vassiliev derivative of Khovanov homology, advancing the understanding of singular links.

Findings

Crux complexes are small for some links, simplifying computations.

The method allows direct calculation of the first derivative of Khovanov homology.

Applied to determine Khovanov complexes of all twist knots universally.

Abstract

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.