Doubled Disks and Satellite Surfaces
arXiv (Cornell University)(2023)
摘要
Conjecturally, a knot is slice if and only if its positive Whitehead double
is slice. We consider an analogue of this conjecture for slice disks in the
four-ball: two slice disks of a knot are smoothly isotopic if and only if their
positive Whitehead doubles are smoothly isotopic. We provide evidence for this
conjecture, using a range of techniques. More generally, we consider when
isotopy obstructions persist under satellite operations. In particular, we show
that obstructions coming from knot Floer homology, Seiberg-Witten theory, and
Khovanov homology often behave well under satellite operations.
We apply these strategies to give a systematic method for constructing vast
numbers of exotic disks in the four-ball, including the first infinite family
of pairwise exotic slice disks. These same techniques are then upgraded to
produce exotic disks that remain exotic after any prescribed number of internal
stabilizations. Finally, we show that the branched double covers of certain
stably-exotic disks become diffeomorphic after a single stabilization with $S^2
\times S^2$, hence stabilizing them yields exotic surfaces that have
diffeomorphic branched covers.
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关键词
disks,surfaces,satellite
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