  Phylogenetic networks can represent evolutionary events that cannot be
described by phylogenetic trees, such as hybridization, introgression, and
lateral gene transfer. Studying phylogenetic networks under a statistical model
of DNA sequence evolution can aid the inference of phylogenetic networks. Most
notably Markov models like the Jukes-Cantor or Kimura-3 model can been employed
to infer a phylogenetic network using phylogenetic invariants. In this article
we determine all quadratic invariants for sunlet networks under the random walk
4-state Markov model, which includes the aforementioned models. Taking toric
fiber products of trees and sunlet networks, we obtain a new class of
invariants for level-1 phylogenetic networks under the same model. Furthermore,
we apply our results to the identifiability problem of a network parameter. In
particular, we prove that our new class of invariants of the studied model is
not sufficient to derive identifiability of quarnets (4-leaf networks).
Moreover, we provide an efficient method that is faster and more reliable than
the state-of-the-art in finding a significant number of invariants for many
level-1 phylogenetic networks.
