相关论文: Lower bounds on the homology of Vietoris-Rips comp…

We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let $Q_n$ be the vertex set of the hypercube graph with $2^n$ vertices, equipped with the shortest path metric.…

We bring in the techniques of independence complexes and the notion of total dominating sets of a graph to bear on the question of the connectivity of the Vietoris-Rips complexes $VR(Q_n; r)$ of an $n$-hypercube graph. We obtain a lower…

We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at scale $3$. We represent the vertices in the hypercube graph $Q_m$ as the collection of all subsets of $[m]=\{1, 2, \ldots, m\}$ and equip $Q_m$ with the metric…

Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known.…

We develop a toric topological framework for studying the cohomology of Vietoris--Rips complexes $VR(Q_n;r)$ of hypercube graphs. Using total domination invariants and spectral methods, we establish general lower bounds on connectivity,…

For a metric space $(X, d)$ and a scale parameter $r \geq 0$, the Vietoris-Rips complex $\mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $\sigma \subseteq X$ is a simplex if and only if diameter of $\sigma$…

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…

For $X$ a metric space and $r>0$ a scale parameter, the Vietoris-Rips complex $VR_<(X;r)$ (resp. $VR_\leq(X;r)$) has $X$ as its vertex set, and a finite subset $\sigma\subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than…

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension…

For a metric space $X$ and $r \geq 0$, the Vietoris-Rips complex $\mathcal{VR}(X;r)$ is a simplicial complex whose simplices are finite subsets of $X$ with diameter at most $r$. Vietoris-Rips complexes have applications in various places,…

We study the concepts of the $\ell_p$-Vietoris-Rips simplicial set and the $\ell_p$-Vietoris-Rips complex of a metric space, where $1\leq p \leq \infty.$ This theory unifies two established theories: for $p=\infty,$ this is the classical…

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…

While the Vietoris-Rips complex is now widely used in both topological data analysis and the theory of hyperbolic groups, many of the fundamental properties of its homology have remained elusive. In this article, we define the Vietoris-Rips…

We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a…

We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\geq 3$ and $k\geq 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the…

Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and…

Given a metric space $(X,d)$, the Vietoris-Rips complex of $X$ at a scale of $r >0$ is a simplicial complex whose simplices are all those finite subsets of $X$ with diameter less than $r$. In this paper, we classify, up to simplicial…

For $X$ a metric space and $r>0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $\sigma \subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than $r$. In ``On…

We prove analogues of classical results for higher homotopy groups and singular homology groups of pseudotopological spaces. Pseudotopological spaces are a generalization of (\v{C}ech) closure spaces which are in turn a generalization of…

We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link…