Related papers: Learning 3-Manifold Triangulations
Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented…
Recently there has been renewed interest in signed distance bound representations due to their unique properties for 3D shape modelling. This is especially the case for deep learning-based bounds. However, it is beneficial to work with…
We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1.…
The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representaci\'on de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a…
Generative models that produce point clouds have emerged as a powerful tool to represent 3D surfaces, and the best current ones rely on learning an ensemble of parametric representations. Unfortunately, they offer no control over the…
This is a survey paper on algorithms for solving problems in 3-dimensional topology. In particular, it discusses Haken's approach to the recognition of the unknot, and recent variations.
We define a trisection of a closed, orientable three dimensional manifold into three handlebodies, and a notion of stabilization for these trisections. Several examples of trisections are described in detail. We define the trisection genus…
Modern 3D semantic instance segmentation approaches predominantly rely on specialized voting mechanisms followed by carefully designed geometric clustering techniques. Building on the successes of recent Transformer-based methods for object…
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…
We present a new local descriptor for 3D shapes, directly applicable to a wide range of shape analysis problems such as point correspondences, semantic segmentation, affordance prediction, and shape-to-scan matching. The descriptor is…
The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving…
Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling…
Computer graphics, 3D computer vision and robotics communities have produced multiple approaches to representing 3D geometry for rendering and reconstruction. These provide trade-offs across fidelity, efficiency and compression…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
Motivated by the algorithmic study of 3-dimensional manifolds, we explore the structural relationship between the JSJ decomposition of a given 3-manifold and its triangulations. Building on work of Bachman, Derby-Talbot and Sedgwick, we…
This article presents a novel method to sampling on manifolds based on the Dirichlet distribution. The proposed strategy allows to completely respect the underlying manifold around which data is observed, and to do massive samplings with…
An important task in image processing and neuroimaging is to extract quantitative information from the acquired images in order to make observations about the presence of disease or markers of development in populations. Having a…
We propose that a simple, Lagrangian 2d $\mathcal{N}=(0, 2)$ duality interface between the 3d $\mathcal{N}=2$ XYZ model and 3d $\mathcal{N}=2$ SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to…
A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on…
We explore the use of a topological manifold, represented as a collection of charts, as the target space of neural network based representation learning tasks. This is achieved by a simple adjustment to the output of an encoder's network…