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Understanding the light ring, one kind fundamental orbit, shall provide uswith novel insight into the astronomical phenomena, such as the ringdown ofbinary merger and shadow of black holes. Recently, topological approach haspreliminarily demonstrated its potential advantages on the properties of thelight rings. However, for the black holes without $\mathbb{Z}_2$ symmetry andextremal spinning black holes are remained to be tested. In this paper, we aimat these two issues. Due to the NUT charge, the Kerr-Newman Taub-NUT solutionhas no $\mathbb{Z}_2$ symmetry. By constructing the corresponding topology forthe non-extremal spinning black holes, we find the topological number keepsunchanged. This indicates that $\mathbb{Z}_2$ symmetry has no influence on thetopological number, while it indeed affects the locations of the light ringsand deviates them off the equatorial plane. For the extremal spinning blackholes, we find its topology is critically dependent of the leading term of thevector's radial component at the zero point of its angular component on theblack hole horizon. The findings state that there exists a topological phasetransition, where the topological number changes, for the prograde light rings.While no phase transition occurs for the retrograde light rings. Our studyuncovers some universal topological properties for the extremal andnon-extremal spinning black holes with or without $\mathbb{Z}_2$ symmetry. Italso has enlightening significance on understanding the light rings in a moregeneral black hole background.

Topology of light rings for extremal and non-extremal Kerr-Newman Taub-NUT black holes without $\mathbb{Z}_2$ symmetry