# Learning to Unknot

arXiv: https://arxiv.org/abs/2010.16263 · HTML: https://mlsanity.com/paper/2010.16263/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2020.

## Built on by (most cited first, 30 of 40)

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- [Machine Learning Clifford invariants of ADE Coxeter elements](https://mlsanity.com/paper/2310.00041/) (2023)
- [Discovery of Optimal Quantum Error Correcting Codes via Reinforcement Learning](https://mlsanity.com/paper/2305.06378/) (2023)
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- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
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- [Untangling Braids with Multi-agent Q-Learning](https://mlsanity.com/paper/2109.14502/) (2021)
- [Learning BPS Spectra and the Gap Conjecture](https://mlsanity.com/paper/2405.09993/) (2024)
- [From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook](https://mlsanity.com/paper/2202.06086/) (2022)
- [An application of neural networks to a problem in knot theory and group theory (untangling braids)](https://mlsanity.com/paper/2206.05373/) (2022)
- [The unknotting number, hard unknot diagrams, and reinforcement learning](https://mlsanity.com/paper/2409.09032/) (2024)
- [Generative AI for Brane Configurations and Coamoeba](https://mlsanity.com/paper/2411.16033/) (2024)
- [Knot So Simple: A Minimalistic Environment for Spatial Reasoning](https://mlsanity.com/paper/2505.18028/) (2025)
- [On knot detection via picture recognition](https://mlsanity.com/paper/2510.06284/) (2025)
- [Intelligent Explorations of the String Theory Landscape](https://mlsanity.com/paper/2204.08073/) (2022)

## Abstract (arXiv)

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.