Problem
Research questions and friction points this paper is trying to address.
slice genus
unknotting number
link invariants
strong slice genus
links
Computations of the slice genus and the unknotting number of links via machine learning
🤖 AI Summary
This study addresses the challenge of estimating upper bounds for topological invariants that are algorithmically difficult to compute, such as the slice genus and unknotting number of links, by introducing machine learning into this domain for the first time. By integrating reinforcement learning with Bayesian optimization to explore unknotting trajectories, and incorporating known lower bounds to approximate exact values, the proposed approach yields new upper bounds and exact values for several link invariants. Furthermore, it successfully reproduces the non-additivity counterexamples established by Brittenham and Hermiller while identifying novel unknotting trajectories. This work establishes a data-driven paradigm for computational topology, effectively overcoming the theoretical limitations inherent in traditional methods.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Reinforcement Learning
Bayesian Optimisation
Slice Genus
Unknotting Number
Link Invariants
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