  In this manuscript, we study the interplay between symmetry and topology with
a focus on the $Z_2$ topological index of 2D/3D topological insulators and
high-order topological insulators. We show that in the presence of either a
two-fold-rotational symmetry or a mirror symmetry, a gauge-invariant quantity
can be defined for arbitrary 1D lines in the Brillouin zone. Such 1D quantities
provide a new pathway to compute the $Z_2$ index of topological insulators. In
contrast to the generic setup, where the $Z_2$ index generally involves 2D
planes in the Brillouin zone with a globally-defined smooth gauge, this 1D
approach only involves some 1D lines in the Brillouin zone without requiring a
global gauge. Such a simplified approach can be used in any time-reversal
invariant insulators with a two-fold crystalline symmetry, which can be found
in 30 of the 32 point groups. In addition, this 1D quantity can be further
generalized to higher-order topological insulators to compute the
magnetoelectric polarization $P_3$.
