Invariants for level-1 phylogenetic networks under the Cavendar-Farris-Neyman model

Phylogenetic networks model evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models used for sequence evolution on trees can also be extended to networks and similar problems, such as determining if the network parameter is identifiable or finding the invariants of the model, can be studied. This paper focuses on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks 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 determine all quadratic invariants for sunlet networks, and conjecture these generate the full sunlet ideal.