3 research outputs found
On Vietoris-Rips complexes of ellipses
For a metric space and a scale parameter, the Vietoris-Rips complex
(resp. ) has as its vertex set, and a finite
subset as a simplex whenever the diameter of is
less than (resp. at most ). 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 of small eccentricity, meaning
. Indeed, we show there are constants such that for
all , we have and , though only one of the two-spheres in is
persistent. Furthermore, we show that for any scale parameter ,
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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On Vietoris-Rips complexes of ellipses
For X a metric space and r>0 a scale parameter, the VietorisâRips complex VR<(X;r) (resp. VRâ¤(X;r)) has X as its vertex set, and a finite subset ĎâX as a simplex whenever the diameter of Ď is less than r (resp. at most r). Though VietorisâRips 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 VietorisâRips complexes of ellipses Y={(x,y)âR2|(x/a)2+y2=1} of small eccentricity, meaning 1<aâ¤2ââ. Indeed, we show there are constants r1<r2 such that for all r1<r<r2, we have VR<(Y;r)âS2 and VRâ¤(Y;r)ââ5S2, though only one of the two-spheres in VRâ¤(Y;r) is persistent. Furthermore, we show that for any scale parameter r1<r<r2, there are arbitrarily dense subsets of the ellipse such that the VietorisâRips complex of the subset is not homotopy equivalent to the VietorisâRips complex of the entire ellipse. As our main tool we link these homotopy types to the structure of infinite cyclic graphs