Ascent concordance

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.

27 pages, 12 figures. Comments welcome. Section 4.2 revised. This version to appear in Algebraic & Geometric Topology

References in corpus (5)

• The Karoubi envelope and Lee's degeneration of Khovanov homology
• Khovanov homology and ribbon concordance
• A deformation of instanton homology for webs
• Cross-sections of unknotted ribbon disks and algebraic curves
• Additional gradings on generalisations of Khovanov homology and invariants of embedded surfaces