The unknotting number, hard unknot diagrams, and reinforcement learning

TL;DR

This paper introduces a reinforcement learning approach to determine the unknotting number of knots, producing a large dataset of complex unknot diagrams and insights into knot properties.

Contribution

The authors developed a reinforcement learning agent that finds minimal unknotting sequences, enabling the determination of unknotting numbers for many knots and generating a large dataset of hard unknot diagrams.

Findings

Successfully determined unknotting numbers for 57,000 knots.

Generated a dataset of 2.6 million hard unknot diagrams.

Discovered examples where crossing changes lead to hyperbolic knots.

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.