Facets in the Vietoris--Rips complexes of hypercubes

Joseph Briggs; Ziqin Feng; Chris Wells

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arXiv · 2408.01288

Facets in the Vietoris--Rips complexes of hypercubes

Joseph Briggs·Ziqin Feng·Chris Wells

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Abstract

In this paper, we investigate the facets of the Vietoris--Rips complex $\mathcal{VR}(Q_n; r)$ where $Q_n$ denotes the $n$-dimensional hypercube. We are particularly interested in those facets which are somehow independent of the dimension $n$. Using Hadamard matrices, we prove that the number of different dimensions of such facets is a super-polynomial function of the scale $r$, assuming that $n$ is sufficiently large. We show also that the $(2r-1)$-th dimensional homology of the complex $\mathcal{VR}(Q_n; r)$ is non-trivial when $n$ is large enough, provided that the Hadamard matrix of order $2r$ exists.

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BibTeX RIS

Joseph Briggs, Ziqin Feng, Chris Wells. 2024-08-02. Facets in the Vietoris--Rips complexes of hypercubes. https://arxiv.org/abs/2408.01288

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