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.
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