Journal article
The unknotting number, hard unknot diagrams, and reinforcement learning
- Abstract:
- We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to determine the unknotting number of 57k knots. We took diagrams of connected sums of such knots with oppositely signed signatures, where the summands were overlaid. The agent has found examples where several of the crossing changes in an unknotting collection of crossings result in hyperbolic knots. Based on this, we have shown that, given knots K and K' that satisfy some mild assumptions, there is a diagram of their connected sum and u(K) + u(K') unknotting crossings such that changing any one of them results in a prime knot. As a by-product, we have obtained a dataset of 2.6 million distinct hard unknot diagrams; most of them under 35 crossings. Assuming the additivity of the unknotting number, we have determined the unknotting number of 43 at most 12-crossing knots for which the unknotting number is unknown.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
Actions
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- Files:
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(Preview, Accepted manuscript, pdf, 649.6KB, Terms of use)
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- Publisher copy:
- 10.1080/10586458.2025.2542174
Authors
- Publisher:
- Taylor & Francis
- Journal:
- Experimental Mathematics More from this journal
- Publication date:
- 2025-08-18
- Acceptance date:
- 2025-07-19
- DOI:
- EISSN:
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1944-950X
- ISSN:
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1058-6458
- Language:
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English
- Keywords:
- Pubs id:
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2246844
- Local pid:
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pubs:2246844
- Deposit date:
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2025-07-19
Terms of use
- Copyright holder:
- Applebaum et al
- Copyright date:
- 2025
- Rights statement:
- © 2025 The Author(s). Published with license by Taylor and Francis Group, LLCThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by theauthor(s) or with their consent.
- Notes:
- The author accepted manuscript (AAM) of this paper has been made available under the University of Oxford's Open Access Publications Policy, and a CC BY public copyright licence has been applied.
- Licence:
- CC Attribution (CC BY)
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