Vietoris--Rips complexes of ellipses at larger scales
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915613067182080 |
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| author | Adams, Henry Carvajal, Julian Rhodes, Jake Turillo, Niccolo Ye, Jingkai Ying, Raymond |
| author_facet | Adams, Henry Carvajal, Julian Rhodes, Jake Turillo, Niccolo Ye, Jingkai Ying, Raymond |
| contents | For $X$ a metric space and $r>0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $σ\subseteq X$ as a simplex whenever the diameter of $σ$ is less than $r$. In ``On Vietoris--Rips complexes of ellipses'', the authors studied the homotopy types of Vietoris--Rips complexes of ellipses $E_a=\{(x,y)\in \mathbb{R}^2~|~(x/a)^2+y^2=1\}$ of small eccentricity, meaning $1<a< \sqrt{2}$, at small scales $r < \frac{4\sqrt{3}a^2}{3a^2+1}$. In this paper, we further investigate the homotopy types that appear at larger scales. In particular, we identify the scale parameters $r$, as a function of the eccentricity $a$, for which the Vietoris--Rips complex $\mathrm{VR}(E_a;r)$ is homotopy equivalent to a $3$-sphere, to a wedge sum of $4$-spheres, or to a $5$-sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_09471 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vietoris--Rips complexes of ellipses at larger scales Adams, Henry Carvajal, Julian Rhodes, Jake Turillo, Niccolo Ye, Jingkai Ying, Raymond Metric Geometry Algebraic Topology 05E45, 55U10, 68R05 For $X$ a metric space and $r>0$, the Vietoris--Rips simplicial complex $\mathrm{VR}(X;r)$ has $X$ as its vertex set, and a finite subset $σ\subseteq X$ as a simplex whenever the diameter of $σ$ is less than $r$. In ``On Vietoris--Rips complexes of ellipses'', the authors studied the homotopy types of Vietoris--Rips complexes of ellipses $E_a=\{(x,y)\in \mathbb{R}^2~|~(x/a)^2+y^2=1\}$ of small eccentricity, meaning $1<a< \sqrt{2}$, at small scales $r < \frac{4\sqrt{3}a^2}{3a^2+1}$. In this paper, we further investigate the homotopy types that appear at larger scales. In particular, we identify the scale parameters $r$, as a function of the eccentricity $a$, for which the Vietoris--Rips complex $\mathrm{VR}(E_a;r)$ is homotopy equivalent to a $3$-sphere, to a wedge sum of $4$-spheres, or to a $5$-sphere. |
| title | Vietoris--Rips complexes of ellipses at larger scales |
| topic | Metric Geometry Algebraic Topology 05E45, 55U10, 68R05 |
| url | https://arxiv.org/abs/2511.09471 |