  Inspired from electronic systems, topological photonics aims to engineer new
optical devices with robust properties. In many cases, the ideas from
topological phases protected by internal symmetries in fermionic systems are
extended to those protected by crystalline symmetries. One such popular
photonic crystal model was proposed by Wu and Hu in 2015 for realizing a
bosonic $\mathbb{Z}_2$ topological crystalline insulator with robust
topological edge states, which led to intense theoretical and experimental
studies. However, rigorous relationship between the bulk topology and edge
properties for this model, which is central to evaluating its advantage over
traditional photonic designs, has never been established. In this work we
revisit the expanded and shrunken honeycomb lattice structures proposed by Wu
and Hu by using topological quantum chemistry tools and show that they are
topologically trivial in the sense that symmetric, localized Wannier functions
can be constructed. We show that the $\mathbb{Z}$ and $\mathbb{Z}_2$ type
classification of the Wu-Hu model are equivalent to the $C_2T$ protected Euler
class and the second Stiefel-Whitney class respectively, with the latter
characterizing the full valence bands of Wu-Hu model indicating only a higher
order topological insulator (HOTI) phase. We show that the Wu-Hu interface
states can be gapped by a uniform topology preserving $C_6$ and $T$ symmetric
perturbation, which demonstrates the trivial nature of the interface. Our
results reveals that topology is not a necessary condition for the reported
helical edge states in many photonics systems and opens new possibilities for
interface engineering that may not be constrained to require topological
designs.
