Vietoris-Rips complexes of torus grids

2025/02/10 by Henry Adams, Adams, Henry, Adenike Yeside Adetowubo +7
Computer Science · Mathematics · #05C69 #05E45 #55N31 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2502.07134

openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the topology of Vietoris--Rips complexes of finite grids on the torus. Let T_n,n be the grid of n× n points on the flat torus S¹× S¹, equipped with the l¹ metric. Let VR(T_n,n;k) be the Vietoris--Rips simplicial complex of this torus grid at scale k≥ 0. For n≥ 7 and small scales 2≤ k≤ (n-1)/(3), the complex VR(T_n,n;k) is homotopy equivalent to the torus. For large scales k≥ 2\lfloor(n)/(2)\rfloor, the complex VR(T_n,n;k) is a simplex and hence contractible. Interesting topology arises over intermediate scales (n-1)/(3)

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