Vietoris–Rips Complexes of Torus Grids

Abstract

In this work, we study the homotopy types and homology of Vietoris–Rips complexes of torus grid graphs. Let T_{n,n} be a torus grid graph consisting of n × n points on T2 equipped with the l1 metric. We first compute the homology of VR(T_{n,n}; k) for 1 ≤ n ≤ 28 and 1 ≤ k ≤ 13. Subsequently, we provide a complete classification of the maximal simplices in VR(T_{3k,3k}; k) for k ≥ 2, VR(T_{3k−1,3k−1}; k) for k ≥ 3, and VR(T_{n,n}; k) when k ≥ 2 and n ≥ 3k. Using these classifications, we establish the homotopy equivalences: VR(T_{3k,3k}; k) ≃ V_{6k^2−1} S2 for k ≥ 2, VR(T_{3k−1,3k−1}; k) ≃ V_{6k−3} S2 ∨ V_{6k−2} S3 for k ≥ 3, and VR(T_{n,n}; k) ≃ T^2 when k ≥ 2 and n ≥ 3k.