  A simple core-shell two-dimensional photonic crystal is studied where the
triangle lattice symmetry and $C_{6v}$ rotation symmetry leads to rich physics
in the study of accidental degeneracy's in photonic bands. We systematically
evaluate different types of accidental nodal points, depending on the
dispersions around them and their topological properties, when the geometry and
permittivity are continuously changed. These accidental nodal points can be the
critical states lying between a topological phase and a normal phase and are
thus important for the study of topological photonic states. In time-reversal
systems, this leads to the photonic quantum spin Hall insulator where the spin
is defined upon the orbital angular momentum for transverse-magnetic
polarization. We study the topological phase transition as well as the
properties of the edge and bulk states and their application potentials in
optics.
