Ascent sliceness

TL;DR

This paper introduces ascent sliceness for virtual knots, analyzing their cobordisms and genus changes via Morse functions, and uses an augmented Khovanov homology to identify ascent slice knots of genus one.

Contribution

It defines ascent sliceness for virtual knots and connects it with an augmented Khovanov homology, providing new tools to study virtual knot cobordisms and genus changes.

Findings

Ascent sliceness is characterized for virtual knots.

Augmented doubled Khovanov homology detects ascent sliceness.

Applicable to slice virtual knots of minimal genus 1.

Abstract

We introduce the notion of ascent sliceness of virtual knots. A representative of a virtual knot is an embedding $ S^1 \hookrightarrow \Sigma_{g} \times I $, for $ \Sigma_g $ a closed connected oriented surface of genus $ g $; the virtual knot represented is slice if there exists a pair consisting of a disc $ D $ and an oriented $ 3 $-manifold $ M $, such that $ D \hookrightarrow M \times I $, $ \partial M = \Sigma_{g} $, and $ \partial D = S^1 $ (the image of the embedding). This definition of sliceness exemplifies that a cobordism of virtual links is a pair consisting of a surface and a $ 3 $-manifold; in addition to analysing the surfaces, as is done in classical knot theory, we may analyse the $ 3 $-manifolds appearing in cobordisms between virtual knots. In particular, consider a Morse function on the $ 3 $-manifold $ M $: away from critical points the level sets are surfaces, and we may ask how the genus of these surfaces changes as we move through the cobordism. Roughly, a slice virtual knot $ K $ with genus-minimal representative $ S^1 \hookrightarrow \Sigma_{g} \times I $ is ascent slice if, given any disc and $ 3 $-manifold pair $ ( D, M ) $ as above, and any Morse function $ f : M \rightarrow I $, the surface $ \Sigma_{g+1} $ appears as a level set of $ f $. We use an augmented version of doubled Khovanov homology to define a property which implies ascent sliceness for slice virtual knots of minimal supporting genus $ 1 $.