  A topological approach to the study of genetic recombination, based on
persistent homology, was introduced by Chan, Carlsson, and Rabad\'an in 2013.
This associates a sequence of signatures called barcodes to genomic data
sampled from an evolutionary history. In this paper, we develop theoretical
foundations for this approach. First, we present a novel formulation of the
underlying inference problem. Specifically, we introduce and study the novelty
profile, a simple, stable statistic of an evolutionary history which not only
counts recombination events but also quantifies how recombination creates
genetic diversity. We propose that the (hitherto implicit) goal of the
topological approach to recombination is the estimation of novelty profiles. We
then study the problem of obtaining a lower bound on the novelty profile using
barcodes. We focus on a low-recombination regime, where the evolutionary
history can be described by a directed acyclic graph called a galled tree,
which differs from a tree only by isolated topological defects. We show that in
this regime, under a complete sampling assumption, the $1^\mathrm{st}$ barcode
yields a lower bound on the novelty profile, and hence on the number of
recombination events. For $i>1$, the $i^{\mathrm{th}}$ barcode is empty. In
addition, we use a stability principle to strengthen these results to ones
which hold for any subsample of an arbitrary evolutionary history. To establish
these results, we describe the topology of the Vietoris--Rips filtrations
arising from evolutionary histories indexed by galled trees. As a step towards
a probabilistic theory, we also show that for a random history indexed by a
fixed galled tree and satisfying biologically reasonable conditions, the
intervals of the $1^{\mathrm{st}}$ barcode are independent random variables.
Using simulations, we explore the sensitivity of these intervals to
recombination.
