  Reticulate evolutionary processes result in phylogenetic histories that
cannot be modeled using a tree topology. Here, we apply methods from
topological data analysis to molecular sequence data with reticulations. Using
a simple example, we demonstrate the correspondence between nontrivial higher
homology and reticulate evolution. We discuss the sensitivity of the standard
filtration and show cases where reticulate evolution can fail to be detected.
We introduce an extension of the standard framework and define the median
complex as a construction to recover signal of the frequency and scale of
reticulate evolution by inferring and imputing putative ancestral states.
Finally, we apply our methods to two datasets from phylogenetics. Our work
expands on earlier ideas of using topology to extract important evolutionary
features from genomic data.
