Learning 3-Manifold Triangulations

We explicitly construct small triangulations for a number of well-known 3-dimensional manifolds and give a brief outline of some aspects of the underlying theory of 3-manifolds and its historical development.

The rising interest in leveraging higher-order interactions present in complex systems has led to a surge in more expressive models exploiting high-order structures in the data, especially in topological deep learning (TDL), which designs neural networks on high-order domains such as simplicial complexes. However, progress in this field is hindered by the scarcity of datasets for benchmarking these architectures. To address this gap, we introduce MANTRA, the first large-scale...

This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several classes of such objects using such structures.

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 learni...

It is important to have effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work that suggests the reality is far different: for an exhaustive census of 81,800,394 one-vertex triangulations that span 1,901 distinct closed orientable 3-manifolds,...

Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data representations. Actually, there is still no manifold learning method that provides a theoretical guarantee of isometry. Inspired by Nash's isometric theorem, we introduce a new concept called isometric immersion learning based on Riemannian geome...

Understanding the topological characteristics of data is important to many areas of research. Recent work has demonstrated that synthetic 4D image-type data can be useful to train 4D convolutional neural network models to see topological features in these data. These models also appear to tolerate the use of image preprocessing techniques where existing topological data analysis techniques such as persistent homology do not. This paper investigates how methods from algebraic ...

In this paper we study the manifolds in the census of "small" 3-manifolds as available in SnapPy. We compare our results with the statistics of random 3-manifolds obtained using the Dunfield Thurston and Rivin models.

Leo Tolstoy opened his monumental novel Anna Karenina with the now famous words: Happy families are all alike; every unhappy family is unhappy in its own way A similar notion also applies to mathematical spaces: Every flat space is alike; every unflat space is unflat in its own way. However, rather than being a source of unhappiness, we will show that the diversity of non-flat spaces provides a rich area of study. The genesis of the so-called big data era and the proliferatio...

This research uses deep learning to estimate the topology of manifolds represented by sparse, unordered point cloud scenes in 3D. A new labelled dataset was synthesised to train neural networks and evaluate their ability to estimate the genus of these manifolds. This data used random homeomorphic deformations to provoke the learning of visual topological features. We demonstrate that deep learning models could extract these features and discuss some advantages over existing t...