Topological Thermodynamics of Black Holes: Revisiting the methods of winding numbers calculation

Abstract

In this paper, the equivalence between two methods for computing winding numbers is established: the approach of $\phi$-mapping topological current and the residue method. The methods are shown to be equivalent when the condition $M'' S' - S'' M' \neq 0$ 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-Nordstr"om 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 in three dimensions, suggesting that it may represent a universal property of cylindrically symmetric black holes in AdS backgrounds.