On Vietoris–Rips complexes of ellipses

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

Michał Adamaszek, Henry Adams, Samadwara Reddy

For (Formula presented.) a metric space and (Formula presented.) a scale parameter, the Vietoris–Rips simplicial complex (Formula presented.) (resp. (Formula presented.)) has (Formula presented.) as its vertex set, and a finite subset (Formula presented.) as a simplex whenever the diameter of (Formula presented.) is less than (Formula presented.) (resp. at most (Formula presented.)). 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 (Formula presented.) of small eccentricity, meaning (Formula presented.). Indeed, we show that there are constants (Formula presented.) such that for all (Formula presented.), we have (Formula presented.) and (Formula presented.), though only one of the two-spheres in (Formula presented.) is persistent. Furthermore, we show that for any scale parameter (Formula presented.), 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.

Originalsprog Engelsk Journal of Topology and Analysis 30 1793-5253 https://doi.org/10.1142/S1793525319500274 Accepteret/In press - 2019

ID: 196738887