Revisiting thermodynamic topologies of black holes

In the generalized off-shell free energy landscape, black holes can be treated as thermodynamic topological defects. The local properties of spacetime reflected by winding numbers at defects, while global nature classified number which is sum all numbers. We propose that calculated via residues isolated one-order pole points characterized functions constructed from energy. Using residue method, we show topologies divided into three classes with being -1, 0, and 1, respectively, consistent results obtained in [Phys. Rev. Lett. 129, 191101 (2022)] using current method. Moreover, point out standard defect points, generation annihilation critical distinguished coefficients Laurent series function those singular points.