Schwarzschild–Letelier Spacetime Surrounded by a King Dark Matter Halo: Geodesic, Shadow, and Thermodynamics

We investigate a static and spherically symmetric Schwarzschild–Letelier Black Hole immersed in a King Dark Matter Halo and analyze how the combined effects of the cloud of strings and the dark-matter environment modify the spacetime geometry, particle dynamics, and thermodynamic behavior of the black hole. Particular attention is devoted to the motion of both massless photons and massive test particles in this black hole background. In the geodesic analysis, we derive the effective potential and study the properties of circular photon orbits, the associated black-hole shadow radius, and the innermost stable circular orbit (ISCO), highlighting the role played by the cloud of strings parameter and the King dark-matter halo parameters in shifting the orbital structure relative to the standard Schwarzschild case. To further characterize the spacetime from a topological perspective, we investigate the unstable circular null orbit using a normalized vector field constructed within the framework of Duan’s ϕ-Mapping Topological Current Theory. Through this method, we identify the corresponding topological charge and examine the relation between the photon sphere and the underlying topological structure of the black-hole configuration. In addition, we explore the thermodynamic properties of the system by computing the Hawking temperature, entropy, Helmholtz free energy, and heat capacity, thereby analyzing the black hole’s local and global thermodynamic stability. The influence of the surrounding dark-matter halo and cloud of strings on the phase structure and thermal behavior is discussed in detail. We further study the thermodynamic topology of the system via the off-shell free-energy formalism, which provides insight into possible thermodynamic phase transitions and the topological classification of black-hole states. Our analysis demonstrates that the combined effects of the cloud of strings and the King dark-matter halo significantly modify the horizon structure, geodesic dynamics, shadow characteristics, and thermodynamic properties of the black hole when compared with the standard Schwarzschild solution.