  The topological nature of nodal lines in a three-band system can be described
by non-Abelian topological charges called quaternion numbers. Due to the gauge
freedom of the eigenstates, the sign of the non-Abelian topological charges can
be flipped by performing the gauge transformation, i.e., choosing a different
basis of eigenstates. However, the sign flipping has not been explicitly shown
in realistic systems such as phononic and photonic topological insulators.
Here, we elaborate the sign freedom by visualizing the numerically calculated
topological charges in phononic and photonic topological insulators. In doing
so, we employ a common reference point method for multiple nodal lines to
confirm that the sign flipping does not cause any issue in building the
quaternion group.
