Virtual knot homology and concordance

Abstract

We construct and investigate the properties of a new extension of Khovanov homology to virtual links, known as doubled Khovanov homology. We describe a perturbation of doubled Khovanov homology, analogous to Lee homology, and produce a doubled Rasmussen invariant; we use it obtain a number of results regarding virtual knot and link concordance. For instance, we demonstrate that the doubled Rasmussen invariant can obstruct the existance of a concordance between a virtual knot and a classical knot (i.e. a knot in the three-sphere ). Kawamura and independently Lobb defned easily-computable bounds on the Rasmussen invariant of classical knots; we generalise these bounds to both the doubled Rasmussen invariant and to a distinct concordance invariant known as the virtual Rasmussen invariant, due to Dye, Kaestner, and Kaufman. We use the new bounds to compute or estimate the slice genus of all virtual knots of 4 classical crossings or less. Finally, we use doubled Khovanov homology as a framework to construct a homology theory of links in thickened surfaces (objects closely related to, but distinct from, virtual links). This homology theory of links in thickened surfaces feeds back to the study of virtual knot concordance, as we are able to use it to investigate a refnement of the notion of sliceness of virtual knots.