Facets in the Vietoris--Rips complexes of hypercubes

2024/08/02 by Briggs, Joseph, Feng, Ziqin, Wells, Chris · 1 citation
#Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.01288

Abstract

In this paper, we investigate the facets of the Vietoris--Rips complex 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 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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