  The Fano resonance with a distinctive ultra-sharp, asymmetric line shape and
high quality factor Q, is a widely occurring phenomena that has a large variety
of optical, plasmonic and microwave manifestations. In this paper, we explore
the characteristic robustness of the Fano resonance mode, which is
topologically protected by manufacturing band inversion induced by breaking the
mirror symmetry of a two-dimensional honeycomb photonic crystal (HPC),
associated with C_6 point group symmetry. So the dark and bright topological
edge modes appear in the band gap made by opening of the Dirac cone.
Destructive and constructive interference of the dark and bright modes leads to
the asymmetric line shape of the Fano resonance. The Fano resonance mode which
is very sensitive to the environmental and geometrical perturbations, can be
applied to sensor design. Here we demonstrate that the topological Fano
resonance mode preserves its asymmetric, ultra-sharp line shape in the presence
of the disorder, defects and cavities, and this has useful optical device
applications such as in low threshold lasers, and extremely precise
interferometers
