Ascent concordance

Rushworth, William

doi:10.2140/agt.2021.21.3073

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Ascent concordance

William Rushworth

Algebraic & Geometric Topology 21 (2021) 3073–3106

DOI: 10.2140/agt.2021.21.3073

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 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.

Keywords

link concordance, links in thickened surfaces, slice–ribbon conjecture

Mathematical Subject Classification

Primary: 57K10, 57K18, 57Q60

References

Publication

Received: 18 May 2020

Revised: 31 July 2020

Accepted: 12 October 2020

Published: 22 November 2021

Authors

William Rushworth
Department of Mathematics Syracuse University Syracuse United States
https://wrushworth.net

Volume 21, issue 6 (2021)