Searching for ribbons with machine learning

Creators: Gukov, Sergei^{1, 2}; Halverson, James^{3, 4}; Manolescu, Ciprian⁵; Ruehle, Fabian^{4, 3}

1. Dublin Institute For Advanced Studies
2. California Institute of Technology
3. Northeastern University
4. AI Institute for Artificial Intelligence and Fundamental Interactions
5. Stanford University

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.

Copyright and License

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Acknowledgement

We would like to thank Nathan Dunfield, Sherry Gong, Mark Hughes, and Lisa Piccirillo for helpful discussions during the preparation of this work. The code for attaching a band using the dual graph of the knot is based on previous work by Gong [Gon].

SG and CM are supported by a Simons Collaboration Grant on New Structures in Low-Dimensional Topology. CM is also supported by a Simons Investigator Award, and the NSF Grant DMS-2003488. SG is also partially supported by the NSF Grant DMS-1664227. JH and FR are supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions). JH is also supported by NSF CAREER Grant PHY-1848089. FR is also supported by NSF Grant PHY-2210333 and startup funding from Northeastern University.

Data Availability

The data cannot be made publicly available upon publication because the cost of preparing, depositing and hosting the data would be prohibitive within the terms of this research project. The data that support the findings of this study are available upon reasonable request from the authors. https://github.com/ruehlef/ribbon.

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