  In phylogenetics, it is important for the phylogenetic network model
parameters to be identifiable so that the evolutionary histories of a group of
species can be consistently inferred. However, as the complexity of the
phylogenetic network models grows, the identifiability of network models
becomes increasingly difficult to analyze. As an attempt to analyze the
identifiability of network models, we check whether two networks are
distinguishable. In this paper, we specifically study the distinguishability of
phylogenetic network models associated with level-2 networks. Using an
algebraic approach, namely using discrete Fourier transformation, we present
some results on the distinguishability of some level-2 networks, which
generalize earlier work on the distinguishability of level-1 networks. In
particular, we study simple and semisimple level-2 networks. Simple and
semisimple level-2 networks can be thought as generalizations of level-1 sunlet
and cycle networks, respectively. Moreover, we also compare the varieties
associated with semisimple level-2 and cycle networks.
