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

Audrey Lindsay, Fabian Ruehle

TL;DR

This paper investigates how neural networks learn knot invariants across multiple knot representations, finding that braid-based encodings generally yield higher predictive accuracy, especially for diagrammatic and hyperbolic invariants. Using KnotInfo/Snappy data up to 13 crossings, it trains 3-layer FFNNs with a systematic hyperparameter box search, and introduces gradient-saliency cosine similarity and joint misclassification scores to quantify neural similarity across invariants. Hyperbolic invariants (Vol, Lon, Mer) are among the easiest to predict, while topological invariants pose greater challenges, with the Arf invariant remaining unlearned under the tested setups. The work identifies strong relationships among invariants (notably $s$, $\tau$, $\sigma$, and $g_4$) and demonstrates how representation choice and data availability shape learnability, suggesting directions for more efficient invariant computation and extended analyses with larger datasets and advanced architectures.

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.

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

TL;DR

This paper investigates how neural networks learn knot invariants across multiple knot representations, finding that braid-based encodings generally yield higher predictive accuracy, especially for diagrammatic and hyperbolic invariants. Using KnotInfo/Snappy data up to 13 crossings, it trains 3-layer FFNNs with a systematic hyperparameter box search, and introduces gradient-saliency cosine similarity and joint misclassification scores to quantify neural similarity across invariants. Hyperbolic invariants (Vol, Lon, Mer) are among the easiest to predict, while topological invariants pose greater challenges, with the Arf invariant remaining unlearned under the tested setups. The work identifies strong relationships among invariants (notably , , , and ) and demonstrates how representation choice and data availability shape learnability, suggesting directions for more efficient invariant computation and extended analyses with larger datasets and advanced architectures.

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.

Paper Structure

This paper contains 11 sections, 8 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Knot Representations
  4. Knot Invariants
  5. Neural Networks
  6. Results
  7. Accuracies
  8. Training dynamics
  9. Similarity in NNs across knot invariants
  10. Dependence on hyperparameters and architecture
  11. Conclusion

Figures (3)

  • Figure 1: Accuracies of knot representation predictions for different knot invariants.
  • Figure 2: Effect of weight decay (wd) and dropout (p) on train and test losses for G4D across different knot representations. Weight decay or early stopping are most effective in preventing over-fitting.
  • Figure 3: Correlation among neural networks for different knot invariants.