# The unknotting number, hard unknot diagrams, and reinforcement learning

arXiv: https://arxiv.org/abs/2409.09032 · HTML: https://mlsanity.com/paper/2409.09032/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 5 of its 35 references that are in the map.
- First posted on arXiv 13 Sep 2024.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 5 of 5)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures](https://mlsanity.com/paper/1802.01561/) (2018)
- [IQ-Learn: Inverse soft-Q Learning for Imitation](https://mlsanity.com/paper/2106.12142/) (2021)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)

## Built on by (most cited first, 2 of 2)

- [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https://mlsanity.com/paper/2412.15184/) (2024)
- [RL unknotter, hard unknots and unknotting number](https://mlsanity.com/paper/2603.07955/) (2026)

## Abstract (arXiv)

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.