  A crucial step in the analysis of persistent homology is the transformation
of data into an appropriate topological object (in our case, a simplicial
complex). Modern packages for persistent homology often construct
Vietoris--Rips or other distance-based simplicial complexes on point clouds
because they are relatively easy to compute. We investigate alternative methods
of constructing these complexes and the effects of making associated choices
during simplicial-complex construction on the output of persistent-homology
algorithms. We present two new methods for constructing simplicial complexes
from two-dimensional geospatial data (such as maps). We apply these methods to
a California precinct-level voting data set, demonstrating that our new
constructions can capture geometric characteristics that are missed by
distance-based constructions. Our new constructions can thus yield more
interpretable persistence modules and barcodes for geospatial data. In
particular, they are able to distinguish short-persistence features that occur
only for a narrow range of distance scales (e.g., voting behaviors in densely
populated cities) from short-persistence noise by incorporating information
about other spatial relationships between precincts.
