On Vietoris–Rips complexes of ellipses

Abstract

For [Formula: see text] a metric space and [Formula: see text] a scale parameter, the Vietoris–Rips simplicial complex [Formula: see text] (resp. [Formula: see text]) has [Formula: see text] as its vertex set, and a finite subset [Formula: see text] as a simplex whenever the diameter of [Formula: see text] is less than [Formula: see text] (resp. at most [Formula: see text]). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev [13,16], they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses [Formula: see text] of small eccentricity, meaning [Formula: see text]. Indeed, we show that there are constants [Formula: see text] such that for all [Formula: see text], we have [Formula: see text] and [Formula: see text], though only one of the two-spheres in [Formula: see text] is persistent. Furthermore, we show that for any scale parameter [Formula: see text], there are arbitrarily dense subsets of the ellipse such that the Vietoris–Rips complex of the subset is not homotopy equivalent to the Vietoris–Rips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs.