On Vietoris-Rips Complexes of Ellipses

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 a finite subset $\sigma \subseteq X$ as a simplex whenever the diameter of $\sigma$ is less than $r$ (resp. at most $r$). Though Vietoris–Rips complexes have been studied at small choices of scale by Hausmann and Latschev [12, 14], they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris–Rips complexes of ellipses $Y = \{(x, y) ∈ \mathbb{R}^2|(x/a)^2 + y^2 = 1\}$ of small eccentricity, meaning $1 < a ≤ \sqrt{2}$. Indeed, we show there are constants $r_1 < r_2$ such that for all $r_1 < r < r_2$, we have $VR_<(Y;r) \simeq S^2$ and $VR≤(Y;r) \simeq \bigvee^5 S^2$, though only one of the two-spheres in $VR_≤(Y;r)$ is persistent. Furthermore, we show that for any scale parameter $r_1 < r < r_2$, 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.