Volume 21, issue 6 (2021)

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Ascent concordance

William Rushworth

Algebraic & Geometric Topology 21 (2021) 3073–3106

A cobordism between links in thickened surfaces consists of a surface S and a 3–manifold M with S properly embedded in M × 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 realized 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.

link concordance, links in thickened surfaces, slice–ribbon conjecture
Mathematical Subject Classification
Primary: 57K10, 57K18, 57Q60
Received: 18 May 2020
Revised: 31 July 2020
Accepted: 12 October 2020
Published: 22 November 2021
William Rushworth
Department of Mathematics
Syracuse University
United States