  Reconstructing the evolutionary past of a family of genes is an important
aspect of many genomic studies. To help with this, simple operations on a set
of sequences called orthology relations may be employed. In addition to being
interesting from a practical point of view they are also attractive from a
theoretical perspective in that e. g. a characterization is known for when such
a relation is representable by a certain type of phylogenetic tree. For an
orthology relation inferred from real biological data it is however generally
too much to hope for that it satisfies that characterization. Rather than
trying to correct the data in some way or another which has its own drawbacks,
as an alternative, we propose to represent an orthology relation $\delta$ in
terms of a structure more general than a phylogenetic tree called a
phylogenetic network. To compute such a network in the form of a level-1
representation for $\delta$, we introduce the novel {\sc Network-Popping}
algorithm which has several attractive properties. In addition, we characterize
orthology relations $\delta$ on some set $X$ that have a level-1 representation
in terms of eight natural properties for $\delta$ as well as in terms for
level-1 representations of orthology relations on certain subsets of $X$.
