[Paper Review] Searching for ribbons with machine learning

Sergei Gukov, James Halverson+2 more|arXiv (Cornell University)|Apr 18, 2023

Geometric and Algebraic Topology4 citations

TL;DR

This paper applies Bayesian optimization and reinforcement learning to determine whether knots bound ribbon disks—a key problem in low-dimensional topology with implications for the smooth 4-dimensional Poincaré conjecture. The authors successfully identify 1,705 knots with up to 14 crossings as ribbon, and rule out 843 potential counterexamples to the conjecture by showing both knots in each pair are ribbon.

ABSTRACT

We apply Bayesian optimization and reinforcement learning to a problem in topology: the question of when a knot bounds a ribbon disk. This question is relevant in an approach to disproving the four-dimensional smooth Poincaré conjecture; using our programs, we rule out many potential counterexamples to the conjecture. We also show that the programs are successful in detecting many ribbon knots in the range of up to 70 crossings.

Motivation & Objective

• To develop machine learning methods that produce mathematically rigorous certificates for ribbon knots, rather than probabilistic approximations.
• To address the open problem of determining whether a knot bounds a ribbon disk, a necessary condition for sliceness.
• To eliminate potential counterexamples to the smooth 4-dimensional Poincaré conjecture by proving that certain knot pairs are ribbon.
• To improve upon existing methods by using reinforcement learning and Bayesian optimization to search for sequences of band moves and Reidemeister moves.
• To explore the limitations of current ML approaches in detecting ribbon status for certain complex knots, such as those from the GST and RBG families.

Proposed method

• Employed Bayesian optimization and reinforcement learning agents to search for sequences of four moves—three Reidemeister moves and one band addition—transforming a knot diagram into a trivial link.
• Used a state representation encoding knot diagrams as planar graphs with crossings and bands, enabling the agent to reason over topological transformations.
• Trained agents on three datasets: Ribbon-to-14 (knots up to 14 crossings), Sym (symmetric ribbon knots), and Unsym (asymmetric ribbon knots), to assess performance differences.
• Implemented a verification step where successful move sequences serve as certificates that a knot is ribbon, which can be independently checked.
• Utilized graph neural networks in future work to better capture knot symmetries and improve generalization across isomorphic dual graphs.
• Explored generative modeling of ribbon knots using simulation-based inference, combining slice obstructions with a ribbon verifier to sample from a posterior distribution of ribbon knots.

Experimental results

Research questions

• RQ1 Can machine learning algorithms produce mathematically rigorous certificates for a knot being ribbon, rather than probabilistic predictions?
• RQ2 How effective are Bayesian optimization and reinforcement learning in discovering sequences of band and Reidemeister moves that transform a knot into a trivial link?
• RQ3 Why do certain knot families, such as the symmetric (Sym) and RBG families, remain undetected by current ML agents despite being known to be ribbon?
• RQ4 To what extent does the data distribution of training examples (e.g., Unsym vs. Sym) affect the performance of ML agents in detecting ribbon knots?
• RQ5 Can machine learning be used to generate new, diverse families of ribbon knots that are not captured by existing parametric constructions?

Key findings

• The Bayesian random walk (Bayes RW) agent solved over 99% of Unsym knots within 100 episodes or 5 minutes, and 70% within 10 minutes.
• On the Ribbon-to-14 dataset, the Bayes RW achieved a 100% success rate with a 10-minute time cutoff, with most knots verified in under one second.
• The method successfully reconfirmed that 1,705 knots with up to 14 crossings are ribbon, matching independent results from Dunfield and Gong.
• The Sym dataset proved significantly harder than Unsym, with the Bayes RW solving only ~80% of Sym knots within 500 episodes, indicating structural bias in the data.
• The RL agent trained from a Naive start performed nearly as well as Bayes RW on Unsym knots, but showed no improvement on Sym knots, highlighting data-dependent performance gaps.
• Several known ribbon knots, including $L_{1,1}$ and $L_{2,1}$ from the GST family and some RBG links, remained undetected, suggesting limitations in current move representation and exploration strategies.