Virtual Khovanov homology using cobordisms

TL;DR

This paper extends Khovanov homology to virtual links using cobordisms, providing a computable, semi-local, and combinatorial framework that generalizes classical link homologies and introduces new characteristic two extensions.

Contribution

It introduces a topological complex for virtual links that generalizes classical Khovanov homology and classifies TQFTs for virtual link homologies, including new characteristic two variants.

Findings

Extended Khovanov homology to virtual links

Classified all TQFTs for virtual link homologies

Provided a computable, semi-local framework

Abstract

We extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and other classical link homologies. We show that our construction allows, over rings of characteristic two, extensions with no classical analogon, e.g. Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent ways. Our construction is computable in the sense that one can write a computer program to perform calculations, e.g. we have written a Mathematica based program. Moreover, we give a classification of all unoriented TQFTs which can be used to define virtual link homologies from our topological construction. Furthermore, we prove that our extension is combinatorial and has semi-local properties. We use the semi-local properties to prove an application, i.e. we give a discussion of Lee's degeneration of virtual homology.