Ascent sliceness

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 $isascentsliceif,givenanydiscand$ 3 $-manifoldpair$ ( D, M ) $asabove,andanyMorsefunction$ f : M \rightarrow I $,thesurface$ \Sigma_{g+1} $appearsasalevelsetof$ 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 $1$.