On Vietoris--Rips complexes of hypercube graphs

TL;DR

This paper characterizes the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters, revealing a progression from points to circles to 3-spheres as the scale increases.

Contribution

It provides explicit descriptions of the homotopy types of Vietoris-Rips complexes for hypercube graphs at scales zero, one, and two, including a formula for the number of 3-spheres.

Findings

Vietoris-Rips complex at scale 0 is a set of points.

At scale 1, it is homotopy equivalent to a wedge of circles.

At scale 2, it is homotopy equivalent to a wedge of 3-spheres with a known count.

Abstract

We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs at small scale parameters. In more detail, let $Q_n$ be the vertex set of the hypercube graph with $2^n$ vertices, equipped with the shortest path metric. Equivalently, $Q_n$ is the set of all binary strings of length $n$, equipped with the Hamming distance. The Vietoris-Rips complex of $Q_n$ at scale parameter zero is $2^n$ points, and the Vietoris-Rips complex of $Q_n$ at scale parameter one is the hypercube graph, which is homotopy equivalent to a wedge sum of circles. We show that the Vietoris-Rips complex of $Q_n$ at scale parameter two is homotopy equivalent to a wedge sum of 3-spheres, and furthermore we provide a formula for the number of 3-spheres. Many questions about the Vietoris-Rips complexes of $Q_n$ at larger scale parameters remain open.