Photon trajectory analysis in Lorentz-violating black holes with a cosmic string and cloud strings

This paper examines gravitational lensing in Schwarzschild-like black holes (BHs) within Kalb–Ramond (KR) gravity incorporating cosmic string (CS) and cloud of strings (CoS) topological configurations. We present the first comprehensive analysis of photon deflection angles and magnification in both Schwarzschild-like BHs in KR gravity pierced by CSs (SKRCS) and Schwarzschild-like BHs in KR gravity with CoS (SKRCoS) spacetimes through dual approaches: perturbative expansions yielding approximate solutions and exact elliptic integral formulations providing complete mathematical descriptions. For CS configurations characterized by Lorentz violation (LV) and CS parameters, deflection angles exhibit systematic modifications through composite geometric factors, while CoS geometries demonstrate distinct characteristics through the composite parameter [Formula: see text] that fundamentally alter light propagation. Magnification analysis reveals distinctive critical curve positioning and amplitude scaling relationships enabling observational discrimination between exotic and conventional BH scenarios. Strong field analysis employing the Bozza–Tsukamoto formalism establishes logarithmic divergence coefficients characterizing scaling behavior near photon spheres. We demonstrate asymptotic local flatness of these spacetimes, ensuring validity of lensing calculations. Observational constraints from Solar System precision tests and Event Horizon Telescope data restrict LV parameters within stringent bounds, while galactic-scale observations permit expanded parameter ranges for CS and CoS configurations.