Vietoris-Rips Complexes of Regular Polygons

TL;DR

This paper characterizes the homotopy types and persistent homology of Vietoris-Rips complexes of regular polygons, revealing complex topological structures including spheres of all dimensions as the scale parameter grows.

Contribution

It provides the first complete description of Vietoris-Rips complexes for regular polygons, especially for those with a number of sides as odd double factorials, using cyclic graph theory.

Findings

Homotopy types include spheres of all dimensions.

Persistent homology is characterized up to a scale approaching the polygon's diameter.

Vietoris-Rips complexes of dense subsets can differ in homology from those of the original polygon.

Abstract

Persistent homology has emerged as a novel tool for data analysis in the past two decades. However, there are still very few shapes or even manifolds whose persistent homology barcodes (say of the Vietoris-Rips complex) are fully known. Towards this direction, let $P_n$ be the boundary of a regular polygon in the plane with $n$ sides; we describe the homotopy types of Vietoris-Rips complexes of $P_n$. Indeed, when $n=(k+1)!!$ is an odd double factorial, we provide a complete characterization of the homotopy types and persistent homology of the Vietoris-Rips complexes of $P_n$ up to a scale parameter $r_n$, where $r_n$ approaches the diameter of $P_n$ as $n\to\infty$. Surprisingly, these homotopy types include spheres of all dimensions. Roughly speaking, the number of higher-dimensional spheres appearing is linked to the number of equilateral (but not necessarily equiangular) stars that can be inscribed into $P_n$. As our main tool we use the recently-developed theory of cyclic graphs and winding fractions. Furthermore, we show that the Vietoris-Rips complex of an arbitrarily dense subset of $P_n$ need not be homotopy equivalent to the Vietoris-Rips complex of $P_n$ itself, and indeed, these two complexes can have different homology groups in arbitrarily high dimensions. As an application of our results, we provide a lower bound on the Gromov-Hausdorff distance between $P_n$ and the circle.