  We investigate topological states of two-dimensional (2D) triangular lattices
with multi-orbitals. Tight-binding model calculations of a 2D triangular
lattice based on $\emph{p}_{x}$ and \emph{p}_{y} orbitals exhibit very
interesting doubly degenerate energy points at different positions ($\Gamma$
and K/K$^{\prime}$) in momentum space, with quadratic non-Dirac and linear
Dirac band dispersions, respectively. Counterintuitively, the system shows a
global topologically trivial rather than nontrivial state with consideration of
spin-orbit coupling due to the "destructive interference effect" between the
topological states at the $\Gamma$ and K/K$^{\prime}$ points. The topologically
nontrivial state can emerge by introducing another set of triangular lattices
to the system (bitriangular lattices) due to the breakdown of the interference
effect. With first-principles calculations, we predict an intrinsic Chern
insulating behavior (quantum anomalous Hall effect) in a family of 2D
triangular lattice metal-organic framework of Co(C$_{21}$N$_{3}$H$_{15}$)
(TPyB-Co) from this scheme. Our results provide a different path and
theoretical guidance for the search for and design of new 2D topological
quantum materials.
