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RL unknotter, hard unknots and unknotting number

Anne Dranowski, Yura Kabkov, Daniel Tubbenhauer

TL;DR

A reinforcement learning pipeline for simplifying knot diagrams is developed that learns move proposals and a value heuristic for navigating Reidemeister moves and is tested on ``very hard''unknot diagrams.

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.

RL unknotter, hard unknots and unknotting number

TL;DR

A reinforcement learning pipeline for simplifying knot diagrams is developed that learns move proposals and a value heuristic for navigating Reidemeister moves and is tested on ``very hard''unknot diagrams.

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 where we recover the recently established and surprising upper bound of three for the unknotting number.
Paper Structure (27 sections, 1 theorem, 26 equations)

This paper contains 27 sections, 1 theorem, 26 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. How to read this paper
  5. Background and notation
  6. Diagrams, planar diagram codes, and moves
  7. Why simplification is hard
  8. A minimal RL primer
  9. Benchmarks used in this paper
  10. The reinforcement-learning unknotter
  11. Planar-diagram encodings
  12. Environment: state, actions, and termination
  13. Move set implemented in spherogram
  14. Reward shaping and "blocking"
  15. Policy and training
  16. ...and 12 more sections

Key Result

Proposition 1

(Empirical.) On the $N=385$ very hard diagrams from ApplebaumBlackwellDaviesEdlichJuhaszLackenbyTomasevZheng-RLunknot, evaluated with a fixed budget of $T=500$ macro-steps and repeated $R=10$ times per instance, the unknotter achieves a mean per-run unknotting rate of $p=94.57\%$ with standard devia

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Proposition 1
  • ...and 2 more