  We present the viewpoint of treating one-dimensional band structures as
Riemann surfaces, linking the unique properties of non-Hermiticity to the
geometry and topology of the Riemann surface. Branch cuts and branch points
play a significant role when this viewpoint is applied to both the
open-boundary spectrum and the braiding structure. An open-boundary spectrum is
interpreted as branch cuts connecting certain branch points, and its
consistency with the monodromy representation severely limits its possible
morphology. A braid word for the Brillouin zone can be read off from its
intersections with branch cuts, and its crossing number is given by the winding
number of the discriminant. These results open new avenues to generate
important insights into the physical behaviors of non-Hermitian systems.
