Learning to Unknot

TL;DR

This paper applies NLP techniques and deep learning models to knot theory, specifically for the UNKNOT problem, and introduces reinforcement learning to find unknotting sequences, revealing new insights into knot simplification.

Contribution

It introduces a novel approach combining NLP, deep learning, and reinforcement learning to study and solve the UNKNOT problem using braid representations.

Findings

Transformer architectures outperform fully-connected networks.

Accuracy increases with braid word length.

Reinforcement learning effectively finds unknotting sequences.

Abstract

We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an algorithm to randomly generate $N$-crossing braids and their knot closures and discussing the induced prior on the distribution of knots, we apply binary classification to the UNKNOT decision problem. We find that the Reformer and shared-QK Transformer network architectures outperform fully-connected networks, though all perform well. Perhaps surprisingly, we find that accuracy increases with the length of the braid word, and that the networks learn a direct correlation between the confidence of their predictions and the degree of the Jones polynomial. Finally, we utilize reinforcement learning (RL) to find sequences of Markov moves and braid relations that simplify knots and can identify unknots by explicitly giving the sequence of unknotting actions. Trust region policy optimization (TRPO) performs consistently well for a wide range of crossing numbers and thoroughly outperformed other RL algorithms and random walkers. Studying these actions, we find that braid relations are more useful in simplifying to the unknot than one of the Markov moves.