Topological thermodynamics of black holes: Revisiting the methods of winding numbers calculation

In this paper, the equivalence between two methods for computing winding numbers is established: the approach of ϕ − mapping topological current and the residue method. The methods are shown to be equivalent when the condition M ″ S ′ − S ″ M ′ ≠ 0 holds, while deviations appear when this relation fails, signaling subtle connections between mass M ( r h ), entropy S ( r h ), and topological structure, with r h being the horizon radius. We first verify this equivalence to Schwarzschild and Reissner–Nordström black holes, recovering known classifications and confirming the consistency of our approach with respect to the validity of the above condition. We then extend the analysis to four-dimensional black strings, regarded as cylindrically symmetric black hole solutions in asymptotically AdS spacetimes. Our results show that both neutral and charged black strings possess the same global topological number, W = + 1 , implying that electric charge does not influence their topological classification. This insensitivity to charge, mirrors earlier findings for BTZ black holes, suggesting that it may represent a universal property of cylindrically symmetric black holes in AdS backgrounds.