We study the topology of Vietoris–Rips complexes of finite grids on the torus. Let \(T_{n,n}\) be the grid of \(n\times n\) points on the flat torus \(S^1\times S^1\) , equipped with the \(l^1\) metric. Let \(\textrm{VR}(T_{n,n};k)\) be the Vietoris–Rips simplicial complex of this torus grid at scale \(k\ge 0\) . For \(n\ge 7\) and small scales \(2\le k\le \frac{n-1}{3}\) , the complex \(\textrm{VR}(T_{n,n};k)\) is homotopy equivalent to the torus. For large scales \(k\ge 2\lfloor \frac{n}{2}\rfloor \) , the complex \(\textrm{VR}(T_{n,n};k)\) is a simplex and hence contractible. Interesting topology arises over intermediate scales \(\frac{n-1}{3}<k<2\lfloor \frac{n}{2}\rfloor \) . For example, we prove that \(\textrm{VR}(T_{2n,2n};2n-1)\cong S^{2n^2-1}\) for \(n\ge 2\) , that \(\textrm{VR}(T_{3n,3n};n)\simeq \vee ^{6n^2-1}S^2\) for \(n\ge 2\) , and that \(\textrm{VR}(T_{3n-1,3n-1};n)\simeq \bigvee _{6n-3} S^2\vee \bigvee _{6n-2}S^3\) for \(n\ge 3\) . Based on homology computations, we conjecture that \(\textrm{VR}(T_{n,n};k)\) is homotopy equivalent to a 3-sphere for a countable family of (n, k) pairs, and we prove this for \((n,k)=(7,4)\) .
