A cobordism realizing crossing change on $\mathfrak{sl}_2$ tangle homology and a categorified Vassiliev skein relation

TL;DR

This paper introduces a cobordism-based morphism in Khovanov homology that extends to singular tangles and satisfies categorified Vassiliev skein relations, providing a new invariant framework.

Contribution

It constructs a genus-one cobordism morphism in Khovanov homology that extends to singular tangles and satisfies categorified Vassiliev relations, unifying various homology theories.

Findings

Invariant under double point moves in tangle diagrams

Extension of Khovanov homology to singular tangles

Satisfaction of categorified Vassiliev skein relations

Abstract

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.