  Understanding the light ring, one kind fundamental orbit, shall provide us
with novel insight into the astronomical phenomena, such as the ringdown of
binary merger and shadow of black holes. Recently, topological approach has
preliminarily demonstrated its potential advantages on the properties of the
light rings. However, for the black holes without $\mathbb{Z}_2$ symmetry and
extremal spinning black holes are remained to be tested. In this paper, we aim
at these two issues. Due to the NUT charge, the Kerr-Newman Taub-NUT solution
has no $\mathbb{Z}_2$ symmetry. By constructing the corresponding topology for
the non-extremal spinning black holes, we find the topological number keeps
unchanged. This indicates that $\mathbb{Z}_2$ symmetry has no influence on the
topological number, while it indeed affects the locations of the light rings
and deviates them off the equatorial plane. For the extremal spinning black
holes, we find its topology is critically dependent of the leading term of the
vector's radial component at the zero point of its angular component on the
black hole horizon. The findings state that there exists a topological phase
transition, where the topological number changes, for the prograde light rings.
While no phase transition occurs for the retrograde light rings. Our study
uncovers some universal topological properties for the extremal and
non-extremal spinning black holes with or without $\mathbb{Z}_2$ symmetry. It
also has enlightening significance on understanding the light rings in a more
general black hole background.
