Doubled disks and exotic satellite surfaces
4 affiliations
- 1Department of Mathematics, Stanford University, Stanford, CA, United States
- 2Department of Mathematics and Computer Science, Rutgers University–Newark, Newark, NJ, United States
- 3Mathematical Institute, University of Oxford, Oxford, United Kingdom, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, United Kingdom
- 4Department of Mathematical Sciences, KAIST, Daejeon, South Korea
Abstract
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 S 2 ; hence stabilizing them yields exotic surfaces that have diffeomorphic branched covers.
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BibTeX
@article{guth2026doubled,
title = {{Doubled disks and exotic satellite surfaces}},
author = {Gary Guth and Kyle Hayden and Sungkyung Kang and JungHwan Park},
journal = {Geometry \& Topology},
year = {2026},
month = oct,
doi = {10.2140/gt.2026.30.3293},
url = {https://doi.org/10.2140/gt.2026.30.3293},
note = {CC BY},
}
RIS
TY - JOUR
TI - Doubled disks and exotic satellite surfaces
AU - Gary Guth
AU - Kyle Hayden
AU - Sungkyung Kang
AU - JungHwan Park
PY - 2026
DA - 2026/10/12
JO - Geometry & Topology
DO - 10.2140/gt.2026.30.3293
UR - https://doi.org/10.2140/gt.2026.30.3293
AB - 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 S 2 ; hence stabilizing them yields exotic surfaces that have diffeomorphic branched covers.
SN - 1364-0380
SN - 1465-3060
ER -