On Vietoris-Rips complexes of ellipses

For $X$ a metric space and $r>0$ a scale parameter, the Vietoris-Rips complex $VR_<(X;r)$ (resp. $VR_\leq(X;r)$) 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, they are not well-understood at larger scale parameters. In this paper we investigate the homotopy types of Vietoris-Rips complexes of ellipses $Y=\{(x,y)\in \mathbb{R}^2~|~(x/a)^2+y^2=1\}$ of small eccentricity, meaning $1<a\le\sqrt{2}$. Indeed, we show there are constants $r_1 < r_2$ such that for all $r_1 < r< r_2$, we have $VR_<(X;r)\simeq S^2$ and $VR_\leq(X;r)\simeq \bigvee^5 S^2$, though only one of the two-spheres in $VR_\leq(X;r)$ 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.