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Vietoris--Rips complexes of ellipses at larger scales

Henry Adams, Julian Carvajal, Jake Rhodes, Niccolo Turillo, Jingkai Ye, Raymond Ying

TL;DR

This work extends the understanding of Vietoris--Rips complexes VR(E_a;r) for ellipses with small eccentricity by exploring larger scale parameters r. Building on prior results that VR(E_a;r) is homotopy equivalent to S^1 for small r and S^2 for intermediate r, the authors introduce a side-length function s_a driven by inscribed 5-pointed stars to analyze higher-scale homotopy types, and conjecture a detailed star-geometry structure governing these changes. Under a central conjecture about the 5-pointed stars, they prove conditional theorems identifying scale intervals where VR(E_a;r) is homotopy equivalent to S^3, S^4, or S^5, with possible wedge sums in singular cases and rank-one phenomena in certain homology groups. A key methodological thread is the translation of geometric extremal structures on E_a into the winding fraction framework for cyclic graphs, enabling a AAR-style transfer from 1-skeleton dynamics to higher-dimensional homotopy types. The work combines rigorous elimination-theory arguments (to locate critical eccentricities a_1,a_2), analytic continuity results for the star-dynamics map F, and computational evidence to motivate and justify the proposed conjecture, with open questions about generalizing to more complex star inscribed configurations and higher-dimensional ellipsoids.

Abstract

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.

Vietoris--Rips complexes of ellipses at larger scales

TL;DR

This work extends the understanding of Vietoris--Rips complexes VR(E_a;r) for ellipses with small eccentricity by exploring larger scale parameters r. Building on prior results that VR(E_a;r) is homotopy equivalent to S^1 for small r and S^2 for intermediate r, the authors introduce a side-length function s_a driven by inscribed 5-pointed stars to analyze higher-scale homotopy types, and conjecture a detailed star-geometry structure governing these changes. Under a central conjecture about the 5-pointed stars, they prove conditional theorems identifying scale intervals where VR(E_a;r) is homotopy equivalent to S^3, S^4, or S^5, with possible wedge sums in singular cases and rank-one phenomena in certain homology groups. A key methodological thread is the translation of geometric extremal structures on E_a into the winding fraction framework for cyclic graphs, enabling a AAR-style transfer from 1-skeleton dynamics to higher-dimensional homotopy types. The work combines rigorous elimination-theory arguments (to locate critical eccentricities a_1,a_2), analytic continuity results for the star-dynamics map F, and computational evidence to motivate and justify the proposed conjecture, with open questions about generalizing to more complex star inscribed configurations and higher-dimensional ellipsoids.

Abstract

For a metric space and , the Vietoris--Rips simplicial complex has as its vertex set, and a finite subset as a simplex whenever the diameter of is less than . In ``On Vietoris--Rips complexes of ellipses'', the authors studied the homotopy types of Vietoris--Rips complexes of ellipses of small eccentricity, meaning , at small scales . In this paper, we further investigate the homotopy types that appear at larger scales. In particular, we identify the scale parameters , as a function of the eccentricity , for which the Vietoris--Rips complex is homotopy equivalent to a -sphere, to a wedge sum of -spheres, or to a -sphere.

Paper Structure

This paper contains 12 sections, 16 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. The side-length function and evidence for Conjecture \ref{['conj:main']}
  5. Proof of Theorem \ref{['thm:extrema']}
  6. Proof of Theorem \ref{['thm:main']}
  7. Proof of Theorem \ref{['thm:main']}
  8. Details regarding the singular case
  9. Joint continuity
  10. Definition of $F$
  11. Continuity of $F$
  12. Conclusion

Key Result

Theorem 1.2

There exist two distinct $a_1, a_2 \in (1, \sqrt{2})$ (with $a_1\approx 1.3299$ and $a_2 \approx 1.4123$) such that $s_a$ has at least twenty local minima and twenty local maxima.

Figures (4)

  • Figure 1: (Left) A minimal inscribed star containing $(a,0)$, and (Right) a maximal inscribed star containing $(0,1)$, both found computationally.
  • Figure 2: The homotopy types of $\mathrm{VR}(E_a;r)$ as $r$ varies, for $a_1^- < a <a_1$, $a_1 < a < a_1^+$, $a_2^- < a < a_2$, or $a_2 < a < a_2^+$.
  • Figure 3: (Left) Not a cyclic graph because there is an edge $0 \to 3$ but not $0 \to 2$. (Middle) The cyclic graph $C_8^2$. (Right) This cyclic graph has winding fraction $\frac{1}{5}$.
  • Figure 4: Each plot shows the value of $s_a$ on the portion of the circle in the first quadrant for a different fixed $a$. In each case, 10000 points are sampled. The 1-2-3-4-5 transition is observed in a neighborhood of $a = a_1 \approx 1.3299$. From top left to bottom: Situation 1, $1 < a <a_1^-$; Situation 2, $a_1^- < a < a_1$; Situation 3, $a = a_1$; Situation 4, $a_1 < a < a_1^+$; Situation 5, $a_1^+ \le a \le a_2^-$.

Theorems & Definitions (32)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • ...and 22 more