Learning Topological Invariance

TL;DR

This work proposes a machine-learning framework to learn topological invariance in knot theory by embedding equivalent knots into a common space and using a Transformer to generate new representatives. It employs contrastive and generative learning, with a student–teacher analysis revealing a strong link between learned embeddings and the Goeritz matrix, and it uses embedding-space explorations to test the Jones Unknot Conjecture. The results show robust invariant learning, effective generation within knot classes, and nuanced insights into which classical invariants are captured by the learned representations. The approach offers a general methodology for discovering topology-aware representations and has potential implications for broader problems in three-manifold topology and invariant extraction.

Abstract

Two geometric spaces are in the same topological class if they are related by certain geometric deformations. We propose machine learning methods that automate learning of topological invariance and apply it in the context of knot theory, where two knots are equivalent if they are related by ambient space isotopy. Specifically, given only the knot and no information about its topological invariants, we employ contrastive and generative machine learning techniques to map different representatives of the same knot class to the same point in an embedding vector space. An auto-regressive decoder Transformer network can then generate new representatives from the same knot class. We also describe a student-teacher setup that we use to interpret which known knot invariants are learned by the neural networks to compute the embeddings, and observe a strong correlation with the Goeritz matrix in all setups that we tested. We also develop an approach to resolving the Jones Unknot Conjecture by exploring the vicinity of the embedding space of the Jones polynomial near the locus where the unknots cluster, which we use to generate braid words with simple Jones polynomials.

Figures (12)

• Figure 1: (a) Illustration of sampling in the Jones polynomial embedding space. (b) Different paths sampled between knots, drawn on a cube for better visibility.
• Figure 2: Semi-hard triplet loss and centroid loss for the MLP Networks.
• Figure 3: Semi-hard triplet loss and centroid loss for the CNN Networks.
• Figure 4: Ablation studies for different braid lengths. The $\text{vol}(K)$ test accuracy decreases monotonically with increasing number of scrambles.
• Figure 5: Ablation studies for different numbers of scrambles. The $\text{vol}(K)$ test accuracy does not demonstrate a clear correlation with the braid length.