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The unknotting numbers, hard unknot diagrams, and reinforcement learning

Created by
  • Haebom

Author

Taylor Applebaum, Sam Blackwell, Alex Davies, Thomas Edlich, Andr as Juh asz, Marc Lackenby, Nenad Toma\v{s}ev, Daniel Zheng

Outline

In this paper, we develop a reinforcement learning agent that frequently finds the minimal sequence of unwinding crossovers for knot pictures with up to 200 intersections. This provides an upper bound on the number of unwindings for 57,000 knots. The pictures of the connected-sum knots are taken as having opposite signatures, and the sums are superimposed. The agent finds examples of multiple unwinding crossovers in the set of unwinding crossovers that produce superparabolic knots. Based on this, we show that given knots K and K' satisfying some weak assumptions, there exist their connected-sum pictures and u(K) + u(K') unwinding crossovers, such that changing one of them produces a prime knot. As a byproduct, we obtain a dataset of 2.6 million distinct difficult-to-unwind knot pictures, most of which have less than 35 intersections. Assuming countability of unwindings, we determine the number of unwindings for 43 knots with up to 12 intersections, where the number of unwindings is unknown.

Takeaways, Limitations

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Takeaways:
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We present a reinforcement learning-based algorithm to efficiently compute the upper bound on the number of unties for knots with up to 200 intersections.
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Determine the untieability of 57,000 knots and create a dataset of 2.6 million difficult-to-untie knot illustrations.
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Proof of countability of the number of unravellings for the sum of the connections of two knots under certain conditions and presentation of conditions for generating prime knots.
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Determination of the number of ties for 43 knots whose number of ties is unknown.
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Limitations:
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Since the results were derived by assuming additivity of the number of solutions, the validity of the assumption needs to be verified.
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The performance of the algorithm may decrease as the number of intersections increases.
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There is no guarantee that the developed algorithm will always find the minimum unraveling crossover order for all knots.
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