Ascent sliceness

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 $.