  We study the effects of a synthetic gauge field and pseudospin-orbit
interaction in a stacked two-dimensional ring-network model. The model was
introduced to simulate light propagation in the corresponding ring-resonator
lattice, and is thus completely bosonic. Without these two items, the model
exhibits Floquet-Weyl and Floquet-topological-insulator phases with
topologically gapless and gapped band structures, respectively. The synthetic
magnetic field implemented in the model results in a three-dimensional
Hofstadter-butterfly-type spectrum in a photonic platform. The resulting gaps
are characterization by the winding number of relevant S-matrices together with
the Chern number of the bulk bands. The pseudospin-orbit interaction is defined
as the mixing term between two pseudospin degrees of freedom in the rings,
namely, the clockwise and counter-clockwise modes. It destroys the
Floquet-topological-insulator phases, while the Floquet-Weyl phase with
multiple Weyl points can be preserved by breaking the space-inversion symmetry.
Implementing both the synthetic gauge field and pseudospin-orbit interaction
requires a certain nonreciprocity.
