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 $S^1 \hookrightarrow \Sigma_{g} \times I$ 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 $1$.