On the Learnability of Knot Invariants: Representation, Predictability, and Neural Similarity
This work addresses the problem of automating knot theory predictions for mathematicians and AI researchers, but it is incremental as it builds on existing methods for representation and learning.
The paper investigated how neural networks can predict knot invariants, finding that braid representations generally perform best and that invariants from hyperbolic geometry are easy to learn while topological ones are harder, with the Arf invariant being unlearnable.
We analyze different aspects of neural network predictions of knot invariants. First, we investigate the impact of different knot representations on the prediction of invariants and find that braid representations work in general the best. Second, we study which knot invariants are easy to learn, with invariants derived from hyperbolic geometry and knot diagrams being very easy to learn, while invariants derived from topological or homological data are harder. Predicting the Arf invariant could not be learned for any representation. Third, we propose a cosine similarity score based on gradient saliency vectors, and a joint misclassification score to uncover similarities in neural networks trained to predict related topological invariants.