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Doubled disks and exotic
satellite surfaces
Gary Guth, Kyle Hayden, Sungkyung Kang and JungHwan Park |
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Geometry & Topology 30 (2026) 3293–3332
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Abstract |
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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 ; hence stabilizing them yields exotic surfaces that have diffeomorphic branched covers. |
Keywords
exotic disks, satellite disks, Whitehead double
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Mathematical Subject Classification
Primary: 57K18, 57K41
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References |
Publication
Received: 3 December 2023
Revised: 24 June 2025
Accepted: 18 April 2026
Published: 12 October 2026
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Jianfeng Lin
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Authors |
| Gary Guth | |
| Department of Mathematics Stanford University Stanford, CA United States |
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| Kyle Hayden | |
| Department of Mathematics and
Computer Science Rutgers University–Newark Newark, NJ United States |
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| Sungkyung Kang | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
Department of Pure Mathematics and
Mathematical Statistics University of Cambridge Cambridge United Kingdom |
| JungHwan Park | |
| Department of Mathematical
Sciences KAIST Daejeon South Korea |
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| © 2026 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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