  Persistent homology is a technique recently developed in algebraic and
computational topology well-suited to analysing structure in complex,
high-dimensional data. In this paper, we exposit the theory of persistent
homology from first principles and detail a novel application of this method to
the field of computational linguistics. Using this method, we search for
clusters and other topological features among closely-associated words of the
English language. Furthermore, we compare the clustering abilities of
persistent homology and the commonly-used Markov clustering algorithm and
discuss improvements to basic persistent homology techniques to increase its
clustering efficacy.
