The Vietoris-Rips Complexes of Finite Subsets of an Ellipse of Small Eccentricity

Abstract

For X a metric space and r > 0 a scale parameter, the Vietoris–Rips complex VR<(X; r) (resp. VR≤(X; r)) has X as its vertex set and has a finite σ ⊆ X as a simplex whenever the diameter of σ is less than r (resp. at most r). Though Vietoris–Rips complexes have been studied at small choices of scale, they are not as well-understood at larger scale parameters. In this paper we describe the homotopy types of the Vietoris–Rips complexes of ellipses. Indeed, for Y an ellipse of small eccentricity, we show there are constants r1 < r2 such that for any sufficiently dense subset X of Y, we have that VR<(X; r) will be homotopy equivalent to a wedge sum of two-spheres. Furthermore, we show that there are arbitrarily dense subsets of the ellipse for which the Vietoris–Rips complex of the subset is homotopy equivalent to a wedge sum of arbitrarily many two-spheres, and hence the homotopy types do not converge as subsets become more dense. As our main tool we link these homotopy types to the structure of cyclic graphs.