  Phylogenetic networks can model more complicated evolutionary phenomena that
trees fail to capture such as horizontal gene transfer and hybridization. The
same Markov models that are used to model evolution on trees can also be
extended to networks and similar questions, such as the identifiability of the
network parameter or the invariants of the model, can be asked. In this paper
we focus on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on
level-1 phylogenetic networks. We do this by reducing the problem to finding
invariants of sunlet networks, which are level-1 networks consisting of a
single cycle with leaves at each vertex. We then determine all quadratic
invariants in the sunlet network ideal which we conjecture generate the full
ideal.
