Abstract. For X a metric space and r > 0 a scale parameter, the Vietoris-Rips simplicial 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 [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 Y = {(x, y) ∈ R 2 | (x/a) 2 + y 2 = 1} of small eccentricity, meaning 1 < a ≤ √ 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) S 2 and VR ≤ (Y ; r) 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.
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