The Vietoris-Rips Complexes of Finite Subsets of an Ellipse of Small Eccentricity
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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.
CitationReddy, Samadwara (2017). The Vietoris-Rips Complexes of Finite Subsets of an Ellipse of Small Eccentricity. Honors thesis, Duke University. Retrieved from https://hdl.handle.net/10161/14249.
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