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.

本文将自然语言处理引入到结实理论中，采用编织词表示法，通过随机生成 n 个 n 次数的编织组织讨论 KN 问题。我们发现改进的神经网络算法和增强学习策略优于其他算法。研究表明编织关系是简化 UNKNOT 相对于 Markov 移动的更实用方法。