A cobordism realizing crossing change on [formula omitted] tangle homology and a categorified Vassiliev skein relation

Abstract

We discuss degree-preserving crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we investigate a sum of cobordisms that yields a morphism between complexes of two diagrams related by a change of crossing, which we call the “genus-one morphism.” We prove 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 invariants for singular tangles that extend sl(2) tangle homologies; 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.