Learning to Unknot

The Big Picture

Imagine handing a tangled ball of string to a mathematician and asking: “Is this actually knotted, or could you untangle it into a simple loop if you had enough patience?” This question, the UNKNOT problem, sits at the heart of one of mathematics’ oldest open challenges. Easy to state, fiendishly hard to solve, and connected to the same technology powering your email autocomplete.

Knots appear throughout fundamental science: in protein folding, in the search for possible string theory universes, and in deep unsolved questions about four-dimensional geometry, including the famous smooth Poincaré conjecture. Existing algorithms scale explosively with complexity. Human intuition fails spectacularly for knots with many crossings.

A team from Caltech, Northeastern, CERN, Oxford, and Warsaw took a lateral approach: they taught a neural network to read knots like sentences, then trained a reinforcement learning agent to untangle them step by step. The result classifies knots with high accuracy and can prove a knot is trivial by showing you the moves.

Key Insight: Knots encoded as braid words are formally equivalent to sentences in a language, and transformer neural networks trained on natural language turn out to be very good at detecting the unknot, with accuracy that improves as the knot gets more complex.

How It Works

The key conceptual leap is representing knots as braid words, sequences of symbols like σ₁, σ₂⁻¹, σ₁, σ₂, where each symbol describes how strands cross over or under each other. Close the ends of a braid and you get a knot. A braid word is a sequence of characters drawn from a finite alphabet, which is exactly what language models were built to handle.

The researchers generated N-crossing braids and balanced datasets of unknots and genuine knots, then trained several neural architectures on yes-or-no classification: given this braid word, is it the unknot?

• Fully-connected networks, the baseline, which performed well but not best
• Shared-QK Transformers, attention models where query and key matrices are tied together, reducing the number of parameters
• Reformers, a memory-efficient transformer variant that uses clever indexing shortcuts to handle long sequences without running out of memory

The Reformer and shared-QK Transformer both outperformed the baseline. What’s less obvious: the longer the braid word, the higher the classification accuracy. More crossings should mean a harder problem, but longer braid words carry more redundancy and context, giving the attention mechanism more signal to work with.

The team also found a deep connection between classifier confidence and the Jones polynomial, a knot invariant (a quantity that stays fixed under any deformation, as long as you don’t cut the knot) introduced in 1984. Highly confident predictions correlated strongly with the degree of the Jones polynomial. The network didn’t know about the Jones polynomial. It rediscovered this correlation from braid words alone.

Unknotting as a Game

Classification tells you whether a knot is trivial. Reinforcement learning tells you how to untangle it.

The researchers formulated unknotting as a game. An RL agent observes the current braid and chooses from legal moves: Markov moves (which change the number of strands while preserving the underlying knot) and braid relations (local rewriting rules that simplify crossings without changing the knot type). The reward is simple: simplify the braid, earn points. Unknot it completely, and win.

Among several RL algorithms tested (PPO, A2C, random walkers), Trust Region Policy Optimization (TRPO) clearly dominated. TRPO constrains how much the policy can change per update, preventing catastrophic forgetting and enabling stable learning across a wide range of crossing numbers.

Looking at the agent’s choices revealed something mathematically concrete: braid relations proved far more useful than one of the two Markov moves (stabilization), which was rarely helpful. This is a direct, actionable insight for mathematicians designing knot simplification algorithms.

Unlike the classifier, the RL agent provides a proof. When it successfully unknots a braid, it hands you an explicit, mechanically verifiable sequence of moves, transforming a probabilistic prediction into a mathematical certificate.

Why It Matters

The UNKNOT problem is believed to be decidable but not known to be in NP; its computational complexity remains open. Tools that rapidly classify or simplify knots help mathematicians probe this question. The Jones polynomial connection raises a natural follow-up: if networks are implicitly approximating this invariant, perhaps they can be reverse-engineered to discover new ones.

On the physics side, knots appear in string theory compactifications, lattice gauge theories, and topological quantum computing. The NLP framing also opens a broader door. Other mathematical structures with discrete symbolic representations (Lie algebras, Calabi-Yau manifolds, Diophantine equations) could be treated as languages and fed to transformers. This paper makes a concrete case for that entire research program.

Bottom Line: By treating knots as sentences and unknotting as a game, this team showed that modern AI can both classify and actively solve one of mathematics’ oldest topological puzzles, while uncovering an unexpected connection between neural networks and the Jones polynomial.