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Abstract We exhibit infinitely many ribbon knots, each of which bounds infinitely many pairwise nonisotopic ribbon disks whose exteriors are diffeomorphic. This family provides a positive answer to a stronger version of an old question of Hitt and Sumners. The examples arise from our main result: a classification of fibered, homotopy‐ribbon disks for each generalized square knot up to isotopy. Precisely, we show that each generalized square knot bounds infinitely many pairwise nonisotopic fibered, homotopy‐ribbon disks, all of whose exteriors are diffeomorphic. When , we prove further that infinitely many of these disks are also ribbon; whether the disks are always ribbon is an open problem.
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Searching for ribbons with machine learning
Abstract We apply Bayesian optimization and reinforcement learning to a problem in topology: the question of when a knot bounds a ribbon disk. This question is relevant in an approach to disproving the four-dimensional smooth Poincaré conjecture; using our programs, we rule out many potential counterexamples to the conjecture. We also show that the programs are successful in detecting many ribbon knots in the range of up to 70 crossings.
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- Award ID(s):
- 1848089
- PAR ID:
- 10606794
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Machine Learning: Science and Technology
- Volume:
- 6
- Issue:
- 2
- ISSN:
- 2632-2153
- Format(s):
- Medium: X Size: Article No. 025065
- Size(s):
- Article No. 025065
- Sponsoring Org:
- National Science Foundation
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