# Doubled Khovanov Homology

## TL;DR

This paper introduces doubled Khovanov homology, a new invariant for virtual links, which helps distinguish non-classical links, provides cobordism obstructions, and relates to the odd writhe of virtual knots.

## Contribution

It defines doubled Khovanov homology, extends Lee's perturbation, introduces a doubled Rasmussen invariant, and applies these to virtual knot theory and cobordism obstructions.

## Key findings

- Doubled Khovanov homology detects non-classical virtual links.
- The doubled Rasmussen invariant obstructs sliceness of virtual knots.
- It relates the odd writhe to sliceness and provides new invariants.

## Abstract

We define a homology theory of virtual links built out of the direct sum of the standard Khovanov complex with itself, motivating the name doubled Khovanov homology. We demonstrate that it can be used to show that some virtual links are non-classical, and that it yields a condition on a virtual knot being the connect sum of two unknots. Further, we show that doubled Khovanov homology possesses a perturbation analogous to that defined by Lee in the classical case and define a doubled Rasmussen invariant. This invariant is used to obtain various cobordism obstructions; in particular it is an obstruction to sliceness. Finally, we show that the doubled Rasmussen invariant contains the odd writhe of a virtual knot, and use this to show that knots with non-zero odd writhe are not slice.