Detecting Homeomorphic 3-manifolds via Graph Neural Networks

TL;DR

This work addresses the homeomorphism problem for graph-manifolds by encoding manifolds with plumbing graphs derived from the JSJ decomposition and evaluating whether two graphs yield homeomorphic spaces using Graph Neural Networks. By comparing GEN, GCN, GAT, and NNConv architectures on a large labeled dataset of plumbing-graph pairs, the study demonstrates that polynomial-time GNN-based classifiers can predict homeomorphism with up to 70.7% accuracy, with NNConv+NNConv performing best. The approach trades exactness for efficiency, highlighting how ML can guide topology and high-energy physics inquiries where exact kirby-type moves are computationally expensive. The paper also sketches extensions to Kirby diagrams, four-manifolds, and relationships to $\,widehat{Z}(q)$-invariants, suggesting ML can aid in conjecture formation for topological and quantum-field-theoretic invariants.

Abstract

Motivated by the enumeration of the BPS spectra of certain 3d $\mathcal{N}=2$ supersymmetric quantum field theories, obtained from the compactification of 6d superconformal field theories on three-manifolds, we study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. Utilizing the JSJ decomposition, a unique representation via a plumbing graph is extracted from a graph-manifold. Homeomorphic graph-manifolds are related via a sequence of von Neumann moves on this graph; the algorithmic application of these moves can determine if two graphs correspond to homeomorphic graph-manifolds in super-polynomial time. However, by employing Graph Neural Networks (GNNs), the same problem can be addressed, at the cost of accuracy, in polynomial time. We build a dataset composed of pairs of plumbing graphs, together with a hidden label encoding whether the pair is homeomorphic. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers (GEN, GCN, GAT, NNConv), followed by an aggregation layer and a classification layer. We discuss the strengths and weaknesses of the different GNNs for this homeomorphism problem.

Figures (3)

• Figure 1: Example of plumbing graph $\Gamma$ with the triplet $(e_i,g_i,r_i)$ explicitly written at the vertices and an $\epsilon=\pm 1$ oriented edge.
• Figure 2: Accuracy curve over the number of epochs for the studied GNN models both for the training set, on the left image, and for the validation one on the right.
• Figure 3: Plot of the Cross-Entropy loss function versus the epochs for each of the studied GNN models. Notice that the loss function for the NNConv+NNConv GNN has not stabilized at the end of the training run.