RL unknotter, hard unknots and unknotting number

arXiv stat.ML / 4/7/2026

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Key Points

The paper presents a reinforcement learning pipeline that simplifies knot diagrams by learning move proposals and a value heuristic to guide Reidemeister moves.
The approach is designed to work on arbitrary knots and links, including challenging unknot diagrams identified as “very hard.”
By using diagram inflation, the authors demonstrate the ability to recover a recently established and surprising upper bound of three for the unknotting number of the knot/link form 4_1#9_10.
The work also introduces a self-improving, workbook-driven extension that iteratively strengthens upper bounds for the unknotting number across a list of prime knots.

Abstract

We develop a reinforcement learning pipeline for simplifying knot diagrams. A trained agent learns move proposals and a value heuristic for navigating Reidemeister moves. The pipeline applies to arbitrary knots and links; we test it on ``very hard'' unknot diagrams and, using diagram inflation, on

4_1\#9_{10}

where we recover the recently established and surprising upper bound of three for the unknotting number. In addition, we explain a self-improving workbook-driven extension of the pipeline that systematically improves unknotting number upper bounds on the list of prime knots.