Topological Invariants of Plumbed 3-Manifolds

In this thesis, we deal with topological invariants of plumbed 3-manifolds by means of not only mathematical analysis but also deep machine learning with Graph Neural Networks (GNN). We first introduce a two-variable refinement $\hat{Z}_a(q,t)$ of plumbed 3-manifold invariants $\hat{Z}_a(q)$, which were previously defined for weakly negative definite plumbed 3-manifolds. For plumbed 3-manifolds with two high-valency vertices, we analytically compute the limit by using the explicit integer solutions of quadratic Diophantine equations in two variables. Based on numerical computations of the recovered $\hat{Z}_a(q)$ for plumbings with two high-valency vertices, we propose a conjecture that the recovered $\hat{Z}_a(q)$, if exists, is an invariant for all tree plumbed 3-manifolds. Then we suggest a formula of the $\hat{Z}_a(q,t)$ for the connected sum of plumbed 3-manifolds in terms of those for the components. Next, 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 certain class of 3-manifolds described by plumbing graphs and apply GNNs to 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 also consider reinforcement learning by a GNN to find a sequence of Neumann moves that relates the pair of graphs if the answer is positive.