Related papers: Operations on Metric Thickenings
We introduce a flexible, categorical framework for large-scale geometry that clarifies basic behaviour of the metric Rips filtration and streamlines some constructions in geometric group theory. The paper has two main parts. First, we…
Fix a finite set of points in Euclidean $n$-space $\euc^n$, thought of as a point-cloud sampling of a certain domain $D\subset\euc^n$. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an…
The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…
The fine curve complex of a surface is a simplicial complex whose vertices are essential simple closed curves and whose $k$-simplices are collections of $k+1$ disjoint curves. We prove that the fine curve complex is homotopy equivalent to…
The Vietoris-Rips filtration, the standard filtration on metric data in topological data analysis, is notoriously sensitive to outliers. Sheehy's subdivision-Rips bifiltration $\mathcal{SR}(-)$ is a density-sensitive refinement that is…
We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we…
We identify the smooth metrics $\mc{M}(M)$ on a manifold $M^n$ with the smooth isometric embeddings $f_g: (M,g) \rightarrow (\mb{S}^{\tn}, \tg)$ into a standard sphere of large dimension $\tn=\tn(n)$, and their Palais isotopic deformations,…
Symmetric edge polytopes are a recent and well-studied family of centrally symmetric polytopes arising from graphs. In this paper, we introduce a generalization of this family to arbitrary simplicial complexes. We show how topological…
The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are…
We study the topology of Vietoris--Rips complexes of finite grids on the torus. Let $T_{n,n}$ be the grid of $n\times n$ points on the flat torus $S^1\times S^1$, equipped with the $l^1$ metric. Let $\mathrm{VR}(T_{n,n};k)$ be the…
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
We show that under certain mild conditions, a metric simplicial complex which satisfies the Ptolemy inequality is a CAT(0) space. Ptolemy's inequality is closely related to inversions of metric spaces. For a large class of metric simplicial…
Let $X$ be a compact K\"ahler manifold. Given a big cohomology class $\{\theta\}$, there is a natural equivalence relation on the space of $\theta$-psh functions giving rise to $\mathcal S(X,\theta)$, the space of singularity types of…
We determine the homotopy type of the Vietoris-Rips complexes of the (vertex sets of the) platonic solids. The most interesting case is that the Vietoris-Rips complex of the dodecahedron is a wedge of nine 3-spheres when the parameter is…
Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips…
We survey what is known and unknown about Vietoris-Rips complexes and thickenings of spheres. Afterwards, we show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective…
In this paper, we settle the homotopy properties of the infinity-morphisms of homotopy (bial)-gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with…
In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These {\it multiarc graphs} naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and…
The so called \v{C}ech and Vietoris-Rips simplicial filtrations are designed to capture information about the topological structure of metric datasets. These filtrations are two of the workhorses in the field of topological data analysis.…
We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse…