Machine learning finds new link invariants and unknotting moves
Computations of the slice genus and the unknotting number of links via machine learning
Machine Learning
Summary
Some shapes made by looping and linking circles in 3D space can be very complicated, and mathematicians want to measure their complexity using numbers called invariants. Two important invariants, the slice genus and the unknotting number, are usually very hard to calculate. The authors used AI techniques like reinforcement learning and Bayesian optimization to find new upper limits on these numbers for many such shapes. They combined these results with known lower limits to exactly determine these invariants for many cases, and even found new ways to simplify these shapes.
What this means in practice
- •For mathematical software developers: Incorporate AI-driven heuristic methods to compute topological link invariants more efficiently for software aiding knot theory research.
- •For cryptography engineers: Use computed link invariants as features in cryptographic protocols that rely on knot theory for security primitives.
Authors
Yutong Dai, Oliver Hayman, András Juhász, Ludovico Morellato
Abstract
Links are disjoint unions of circles smoothly embedded in $S^3$. We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.