The Vietoris-Rips Complex for Embedded Lattice Points in the Torus

Abstract

We work towards answering a question posed by Michal Adamaszek and Henry Adams in [1] about the homotopy type for Vietoris-Rips complex of the torus T2 = S1 × S1 with the flat metric by studying a class of finite subsets of the torus: embedded lattice points. Using computational tools, we compute the Betti numbers of some examples of these complexes for the embedded lattices for with arbitrary radius. Based on the computational results, we conjecture about an upper bound for r which ensures that the Vietoris-Rips complex of these embedded lattice points is homotopic to the torus. For some specific r, we have proven this conjecture using discrete Morse theory.