  There has been much recent interest and progress on topological structures of
the non-Hermitian Bloch bands. Here, we study the topological structures of
non-Bloch bands of non-Hermitian multi-band quantum systems under open boundary
conditions, which has received limited attention in prior studies. Using a
continuity criterion and an efficient sub-GBZ algorithm, we establish a
homotopic characterization -- braiding topology, e.g., characterized by the
band's total vorticity -- for open-boundary bands and sub-GBZs. Such
topological identification is robust without topological transition and
emergent degenerate points, such as exceptional points. We further analyze the
transition's impact on bands and spectral flows, including interesting
properties unique to open boundaries, and numerically demonstrate our
conclusions with tight-binding model examples. We unveil a crucial insight that
open-boundary bands interchange their portions after encountering certain
exceptional points. Our results enrich our foundational understanding of
topological characterizations for generic non-Hermitian quantum systems.
