The unknotting number, hard unknot diagrams, and reinforcement learning

2024-09-13

arXiv.org

AI Summary

This study addresses the computational challenge of determining unknotting numbers for knot diagrams with high crossing numbers (≤200) and the generation/identification of “hard unknot diagrams.” We propose a deep reinforcement learning framework based on the Proximal Policy Optimization (PPO) algorithm. Methodologically, we integrate planar diagram (PD) code representations, Reidemeister move action space modeling, symbolic signature constraints for optimization, and large-scale parallel diagram reduction. Key contributions include: (1) systematic computation of unknotting numbers for 57,000 knots, yielding first-time determinations—under the additivity assumption—for 43 previously unknown unknotting numbers of prime knots with ≤12 crossings; (2) construction of the first large-scale hard unknot diagram dataset comprising 2.6 million samples, predominantly with <35 crossings; and (3) the first proof that certain connected sum diagrams exist for which every single crossing change required to unknot yields a prime knot. This work advances the integration of knot invariant theory with artificial intelligence.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Develop RL agent to find minimal unknotting sequences for complex knots
• Determine unknotting numbers for thousands of unknown knot cases
• Generate dataset of hard unknot diagrams with under 35 crossings

Innovation

• Reinforcement learning agent for unknotting diagrams
• Determines unknotting number for 57k knots
• Generates dataset of 2.6M hard unknot diagrams