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≤(X;r)) has X as its vertex set, and a finite subset 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 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)∈R2|(x/a)2+y2=1} of small eccentricity, meaning 1<a≤2–√. Indeed, we show there are constants r1<r2 such that for all r1<r<r2, we have VR<(Y;r)≃S2 and VR≤(Y;r)≃⋁5S2, though only one of the two-spheres in VR≤(Y;r) is persistent. Furthermore, we show that for any scale parameter r1<r