Khovanov Homology, Lee Homology and a Rasmussen Invariant for Virtual Knots

TL;DR

This paper extends Khovanov and Lee homology, along with the Rasmussen invariant, to virtual knots, providing new computational tools and generalizations for virtual knot cobordisms and slice genus.

Contribution

It introduces an alternate formulation of Khovanov homology for virtual knots using cut loci, and generalizes the Rasmussen invariant to virtual knots and cobordisms.

Findings

Generalized Rasmussen invariant for virtual knots

Constructed canonical generators of Khovanov-Lee homology for virtual knots

Provided an obstruction to knot cobordisms in certain 3-manifolds

Abstract

The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies to classical knot and classical knot cobordisms. To do so, we give an alternate formulation for the Manturov definition of Khovanov homology for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius algebra. We then discuss the implications of the maps induced in the aforementioned theory to the universal Frobenius algebra for virtual knots. Next we show how one can apply the Karoubi envelope approach of Bar-Natan and Morrison on abstract link diagrams with cross cuts to construct the canonical generators of the Khovanov-Lee homology. Using these canonical generators we derive a generalization of the Rasmussen invariant for virtual knot cobordisms and furthermore generalize Rasmussen's result on the slice genus for positive knots to the case of positive virtual knots. It should also be noted that this generalization of the Rasmussen invariant provides an easy to compute obstruction to knot cobordisms in $S_g \times I \times I$ in the sense of Turaev.