# Ascent concordance

## TL;DR

This paper demonstrates that in thickened surfaces, low surface complexity does not necessarily imply low 3-manifold complexity in cobordisms, revealing new phenomena in link concordance and providing counterexamples to a Slice-Ribbon conjecture analogue.

## Contribution

It introduces examples of links in thickened surfaces where simple surfaces require complex 3-manifolds for cobordisms, and uses augmented Khovanov homology to detect these cases.

## Key findings

- Existence of links with complex cobordisms despite simple surfaces
- Detection of such links via augmented Khovanov homology
- Counterexamples to an analogue of the Slice-Ribbon conjecture

## Abstract

A cobordism between links in thickened surfaces consists of a surface $ S $ and a $3$-manifold $M $, with $ S $ properly embedded in $ M \times I $. We show that there exist links in thickened surfaces such that if $(S,M) $ is a cobordism between them in which $ S $ is simple, then $ M $ must be complex. That is, there are cases in which low complexity of the surface does not imply low complexity of the $3$-manifold. Specifically, we show that there exist concordant links in thickened surfaces between which a concordance can only be realised by passing through thickenings of higher genus surfaces. We exhibit an infinite family of such links that are detected by an elementary method and other families of links that are not detectable in this way. We investigate an augmented version of Khovanov homology, and use it to detect these families. Such links provide counterexamples to an analogue of the Slice-Ribbon conjecture.