Graph Neural Networks and 3-Dimensional Topology
This work addresses a specific problem in low-dimensional topology for researchers, but it is incremental as it applies existing GNN methods to a new domain without major methodological innovations.
The paper tackles the problem of deciding whether pairs of plumbing graphs yield homeomorphic 3-manifolds using Graph Neural Networks, achieving high accuracy in classification and employing reinforcement learning to find sequences of moves for positive cases.
We test the efficiency of applying Geometric Deep Learning to the problems in low-dimensional topology in a certain simple setting. Specifically, we consider the class of 3-manifolds described by plumbing graphs and use Graph Neural Networks (GNN) for the problem of deciding whether a pair of graphs give homeomorphic 3-manifolds. We use supervised learning to train a GNN that provides the answer to such a question with high accuracy. Moreover, we consider reinforcement learning by a GNN to find a sequence of Neumann moves that relates the pair of graphs if the answer is positive. The setting can be understood as a toy model of the problem of deciding whether a pair of Kirby diagrams give diffeomorphic 3- or 4-manifolds.