Learning 3-Manifold Triangulations

TL;DR

This paper explores the use of machine learning on databases of 3-manifold triangulations, represented by isomorphism signatures, to differentiate manifolds and analyze their structure, revealing new geometric insights.

Contribution

It introduces a novel approach combining machine learning and network analysis on 3-manifold triangulation data to uncover structural properties and relationships.

Findings

Machine learning models can differentiate manifolds based on triangulation signatures.

Gradient saliency highlights key features in the triangulation encoding.

A relation between systole length and Pachner graph size is observed in hyperbolic manifolds.

Abstract

Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $<8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed.