The unknotting number, hard unknot diagrams, and reinforcement learning (2409.09032v1)

Abstract: 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.

• The paper presents a reinforcement learning framework that computes minimal unknotting sequences, providing evidence for the additivity of the unknotting number.
• It introduces a comprehensive dataset of 2.6 million hard unknot diagrams, enhancing our understanding of computational complexity in knot theory.
• The research bridges classical knot theory with modern machine learning, offering practical tools for exploring complex knot invariants and topology.

The Unknotting Number, Hard Unknot Diagrams, and Reinforcement Learning

Introduction

The paper "The Unknotting Number, Hard Unknot Diagrams, and Reinforcement Learning" presents a comprehensive paper of knot theory, focusing on the unknotting number, hard unknot diagrams, and the application of reinforcement learning (RL) techniques to compute unknotting sequences. Knot theory is central to low-dimensional topology, where the unknotting number is a classical, yet intricate knot invariant defined as the minimum number of crossing changes required to transform a knot into an unknot. The researchers leverage modern machine learning paradigms, chiefly reinforcement learning, to explore these classical problems, driven by both theoretical and practical motivations.

Unknotting Number and Knot Diagrams

Definitions and Background

A knot is a smooth embedding of $K: S^1 \hookrightarrow S^3$ and can be represented via projections onto $S^2$, resulting in knot diagrams. Two knot diagrams are equivalent if related by a sequence of Reidemeister moves (R1-R3). The unknotting number $u(K)$ of a knot $K$ is the minimal number of crossing changes needed to transform $K$ into the unknot $U$.

The unknotting number can alternatively be described via crossing arcs and regular homotopies. Calculating $u(K)$ is notoriously difficult due to the lack of an algorithm, partly because $u(K)$ is often smaller than $u(D)$ for any minimal crossing number diagram $D$ of $K$.

Additivity and Connected Sums

One of the long-standing open questions in knot theory is the additivity of the unknotting number under connected sum, stated formally as:

Conjecture: For knots $K$ and $K'$, $u(K \# K') = u(K) + u(K')$.

The authors explored potential counterexamples to this conjecture but instead found substantial evidence supporting it, along with new insights into prime knot configurations and inter-component crossing changes.

Reinforcement Learning in Knot Theory

Methodology

The researchers employed two machine learning paradigms: imitation learning and reinforcement learning, utilizing supervised learning models and the Importance Weighted Actor-Learner Architecture (IMPALA) respectively. The RL agent was designed to find efficient unknotting sequences for diagrams with up to 200 crossings by leveraging features such as the Jones polynomial and Alexander polynomial.

Training and Results

The RL agent was trained on a dataset containing both randomly generated knots and those with known unknotting numbers, including connected sums. The agent demonstrated the capability to determine minimal unknotting sequences and uncovered 43 prime knots with at most 12 crossings whose unknotting numbers were previously unknown, assuming the additivity conjecture holds.

Hard Unknot Diagrams

Definitions and Dataset

A hard unknot diagram is a knot diagram that requires an increase in the number of crossings before it can be reduced to the trivial knot using a sequence of Reidemeister moves. The authors assembled a substantial dataset of 2.6 million distinct hard unknot diagrams, verified through extensive R3 move equivalence checks. This dataset contributes significantly to the paper of unknot detection algorithms and knot complexity.

Implications

The dataset of hard unknot diagrams provides valuable insights into counterexamples to potential polynomial-time unknot detection algorithms. This contribution underpins further research into computational complexity in knot theory.

Theoretical and Practical Implications

The theoretical contributions of this paper include evidence supporting the additivity of the unknotting number and the development of counterexamples to a stronger form of the conjecture. Practically, the reinforcement learning agent provides a powerful tool for knot theorists to explore unknotting sequences in large diagrams efficiently.

Future Developments

Future research could involve further refining RL techniques to enhance performance on even larger and more complex knot diagrams. Additionally, exploring invariants and methods beyond the Jones and Alexander polynomials might unearth new unknotting information. The intersection of machine learning and mathematical theory continues to hold promising potential for deepening our understanding of low-dimensional topology.

Conclusion

This paper bridges classical knot theory with advanced machine learning methodologies. By applying RL to compute unknotting sequences and exploring the properties of hard unknot diagrams, the research advances our computational and theoretical grasp of knot invariants. The results open new pathways for further explorations in both the mathematical and machine learning domains, highlighting the interplay between deep theoretical problems and cutting-edge computational techniques.