" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

One-dimensional persistent homology is arguably the most important and heavily used
computational tool in topological data analysis. Additional information can be extracted from …

We describe the homotopy types of Vietoris–Rips complexes of hypercube graphs at small
scale parameters. In more detail, let Q_n Q n be the vertex set of the hypercube graph with …

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 …

For a metric space and a scale parameter, the Vietoris–Rips complex is a simplicial complex
on vertex set, where a finite set is a simplex if and only if the diameter of is at most. For, let …

Cohomological ideas have recently been injected into persistent homology and have for
example been used for accelerating the calculation of persistence diagrams by the software …

We study notions of persistent homotopy groups of compact metric spaces together with their
stability properties in the Gromov-Hausdorff sense. We pay particular attention to the case of …

Motivated by computational aspects of persistent homology for Vietoris-Rips filtrations, we
generalize a result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of …

Abstract In 1995 Jean-Claude Hausmann proved that a compact Riemannian manifold X is
homotopy equivalent to its Rips complex Rips (X, r) for small values of parameter r. He then …

Persistent homology has been an important tool in topological and geometrical data
analysis to study the shape of data. However, its ability to differentiate between various …