Abstract
A cobordism between links in thickened surfaces consists of a surface and a –manifold with properly embedded in . We show that there exist links in thickened surfaces such that if is a cobordism between them in which is simple, then must be complex. That is, there are cases in which low complexity of the surface does not imply low complexity of the –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.
Citation
William Rushworth. "Ascent concordance." Algebr. Geom. Topol. 21 (6) 3073 - 3106, 2021. https://doi.org/10.2140/agt.2021.21.3073
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