Vietoris-Rips complexes of n-by-n torus grids are shown to be tori, spheres, or wedges of spheres for several infinite families of grid sizes and scales.
Contractible Vietoris-Rips complexes of $\mathbb{Z}^n$
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abstract
We give a new, short proof of a result of Virk, that the Vietoris-Rips complex of the group $\mathbb{Z}^n$ with the standard word metric is contractible at large enough scales. This is inspired by a key observation in Virk's proof, but we use Bestvina-Brady discrete Morse theory to get a very short proof with better bounds. In the course of this, we get a new, general criterion for a metric space to have contractible Vietoris-Rips complexes at large enough scales, which could prove useful in the future.
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Vietoris-Rips complexes of torus grids
Vietoris-Rips complexes of n-by-n torus grids are shown to be tori, spheres, or wedges of spheres for several infinite families of grid sizes and scales.