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

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