  Elementary band representations are the fundamental building blocks of atomic
limit band structures. They have the defining property that at partial filling
they cannot be both gapped and trivial. Here, we give two examples -- one each
in a symmorphic and a non-symmorphic space group -- of elementary band
representations realized with an energy gap. In doing so, we explicitly
construct a counterexample to a claim by Michel and Zak that single-valued
elementary band representations in non-symmorphic space groups with
time-reversal symmetry are connected. For each example, we construct a
topological invariant to explicitly demonstrate that the valence bands are
non-trivial. We discover a new topological invariant: a movable but unremovable
Dirac cone in the "Wilson Hamiltonian" and a bent-Z2 index.
