Ascent concordance

Rushworth, William

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.

Citation

Rushworth, W. (2021). Ascent concordance. Algebraic and Geometric Topology, 21(6), 3073–3106. https://doi.org/10.2140/agt.2021.21.3073

• Journal Article Type: Article
• Acceptance Date: Oct 12, 2020
• Online Publication Date: Nov 22, 2021
• Publication Date: Nov 22, 2021
• Deposit Date: Nov 19, 2022
• Publicly Available Date: Jan 18, 2023
• Journal: Algebraic & Geometric Topology
• Print ISSN: 1472-2747
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 6
• Pages: 3073–3106
• DOI: https://doi.org/10.2140/agt.2021.21.3073
• Keywords: Geometric Topology
• Public URL: https://hull-repository.worktribe.com/output/4127515