On the Learnability of Knot Invariants: Representation, Predictability, and Neural Similarity

TL;DR

This paper investigates how neural networks predict knot invariants, examining the effects of different representations, the learnability of various invariants, and introducing methods to analyze neural similarity in topological predictions.

Contribution

It provides insights into the influence of knot representations on neural prediction accuracy, identifies which invariants are easier to learn, and introduces new similarity measures for neural network analysis.

Findings

Braid representations generally yield better predictions.

Hyperbolic geometry invariants are easier to learn than topological ones.

The Arf invariant remains difficult for neural networks to predict.

Abstract

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.