Learning 3-Manifold Triangulations

TL;DR

This work studies 3-manifold triangulations through the IsoSig encoding, training supervised models to identify which manifold a given triangulation represents and how surgeries affect topology. By generating Pachner graphs and extracting IsoSig datasets, the authors apply neural networks (with gradient saliency) and transformers to classify manifold pairs, and extend analyses to knot complements and Dehn surgeries. Key findings include high classification accuracy for many manifold pairs, interpretable saliency maps highlighting IsoSig subsequences tied to the encoding scheme, and evidence that hyperbolic manifolds exhibit distinctive Pachner-graph growth related to systole length. The results demonstrate a viable, data-driven approach to probing triangulation structure and manifold properties, with implications for speeding up topology computations and guiding future graph-based ML methods.

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.

Figures (21)

• Figure 1: Diagrammatics of the four Pachner moves, including (a) the $2-3$ Pachner move with its inverse $3-2$, and (b) the $1-4$ Pachner move, with its inverse $4-1$. For the $2-3$ move the coloured lines bound a face which is replaced by an edge which then bounds the three tetrahedra inside. For the $1-4$ move a vertex (coloured) is introduced into the centre and then connected to the remaining vertices, splitting the single tetrahedra into four.
• Figure 2: IsoSig length frequency distributions across the deep Pachner graphs generated for the 8 manifolds considered.
• Figure 3: 1-vertex Pachner graphs for the $S^3$ manifold, generated with $\{2-3,3-2\}$ moves up to the respective specified depths, from the initial triangulation with IsoSig: cMcabbgqs. The nodes are labelled with the number of tetrahedra in the respective triangulation.
• Figure 4: The growth rates of the Pachner graphs with depth for the 8 selected manifolds of focus in this work. Plot (a) shows how number of nodes increases with depth, whilst (b) shows the respective decreases in graph density.
• Figure 5: Network analysis of the orientable cusped census Pachner graphs (where number of initial tetrahedra $<8$). Analysis considers building the Pachner graphs to depth 3 with either only the $\{1-4, 4-1\}$ moves (denoted 3_14), or only the $\{2-3, 3-2\}$ (denoted 3_23) moves, plotting histograms of: (a) number of nodes in the graph; (b) graph density; and (c) the graph degree distribution averaged across the census. Note the degrees take integer values and the vertical lines are offsetted for visibility.