  Algebraic techniques in phylogenetics have historically been successful at
proving identifiability results and have also led to novel reconstruction
algorithms. In this paper, we study the ideal of phylogenetic invariants of the
Cavender-Farris-Neyman (CFN) model on a phylogenetic network with the goal of
providing a description of the invariants which is useful for network
inference. It was previously shown that to characterize the invariants of any
level-1 network, it suffices to understand all sunlet networks, which are those
consisting of a single cycle with a leaf adjacent to each cycle vertex. We show
that the parameterization of an affine open patch of the CFN sunlet model,
which intersects the probability simplex factors through the space of
skew-symmetric matrices via Pfaffians. We then show that this affine patch is
isomorphic to a determinantal variety and give an explicit Gr{\"o}bner basis
for the associated ideal, which involves only polynomially many coordinates.
Lastly, we show that sunlet networks with at least 6 leaves are identifiable
using only these polynomials and run extensive simulations, which show that
these polynomials can be used to accurately infer the correct network from DNA
sequence data.
