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Abstract We consider the set of connected surfaces in the 4‐ball with boundary a fixed knot in the 3‐sphere. We define the stabilization distance between two surfaces as the minimal such that we can get from one to the other using stabilizations and destabilizations through surfaces of genus at most . Similarly, we consider a double‐point distance between two surfaces of the same genus that is the minimum over all regular homotopies connecting the two surfaces of the maximal number of double points appearing in the homotopy. To many of the concordance invariants defined using Heegaard Floer homology, we construct an analogous invariant for a pair of surfaces. We show that these give lower bounds on the stabilization distance and the double‐point distance. We compute our invariants for some pairs of deform‐spun slice disks by proving a trace formula on the full infinity knot Floer complex, and by determining the action on knot Floer homology of an automorphism of the connected sum of a knot with itself that swaps the two summands. We use our invariants to find pairs of slice disks with arbitrarily large distance with respect to many of the metrics we consider in this paper. We also answer a slice‐disk analog of Problem 1.105 (B) from Kirby's problem list by showing the existence of non‐0‐cobordant slice disks.
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This content will become publicly available on November 22, 2025
For exotic surfaces with boundary, one stabilization is not enough
Works of Hosokawa–Kawauchi (1979) and Baykur–Sunukjian (2016) show that homologous surfaces in a 4-manifold become isotopic after a finite number of internal stabilizations, i.e., attaching tubes to the surfaces. A natural question is how many stabilizations are needed before the surfaces become isotopic. In particular, given an exotic pair of surfaces, is a single stabilization always enough to make the pair smoothly isotopic? We answer this question by studying how the stabilization distance between surfaces with boundary changes with respect to satellite operations. Using a range of Floer theoretic techniques, we show that there are exotic disks in the 4-ball which have arbitrarily large stabilization distance, giving the first examples of exotic behavior in the 4-ball for which “one is not enough”.
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- Award ID(s):
- 2204214
- PAR ID:
- 10591494
- Publisher / Repository:
- European Mathematical Society
- Date Published:
- Journal Name:
- Journal of the European Mathematical Society
- ISSN:
- 1435-9855
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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