Dimensions of nonzero reduced homology

Determine the dimensions i in which the reduced homology groups of VR(T_{n,n},k) are nonzero for arbitrary torus-grid parameters n and k.

Background

The computational tables display homology in many different dimensions as the grid size and scale vary, while the paper establishes only selected infinite families of homotopy types. A general characterization of the degrees supporting reduced homology remains unresolved.