The Vietoris-Rips complexes of a circle

TL;DR

This paper investigates the changing homotopy types of Vietoris-Rips complexes of a circle as the distance parameter varies, revealing a sequence of topological shapes from the circle to a point, and introduces a new invariant called the winding fraction.

Contribution

The paper introduces the winding fraction invariant to classify homotopy types of Vietoris-Rips complexes of the circle for all distance parameters, extending understanding beyond small r.

Findings

Vietoris-Rips complexes of a circle change homotopy types as r increases, cycling through spheres of odd dimension.

The winding fraction classifies the homotopy type of the complex for any subset of the circle.

Both Vietoris-Rips and Cech complexes exhibit a sequence of homotopy types from circle to contractible as r increases.

Abstract

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 r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. As our main tool we introduce a directed graph invariant, the winding fraction, which in some sense is dual to the circular chromatic number. Using the winding fraction we classify the homotopy types of the Vietoris-Rips complex of an arbitrary (possibly infinite) subset of the circle, and we study the expected homotopy type of the Vietoris-Rips complex of a uniformly random sample from the circle. Moreover, we show that as the distance parameter increases, the ambient Cech complex of the circle also obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible.