Multimodal signatures of asymptotic (A)dS Kalb-Ramond black holes: Constraints through the shadow, weak deflection angle, and topological photon spheres

This study explores novel static, neutral black hole solutions within Kalb-Ramond (KR) gravity in asymptotically (anti-)de Sitter [(A)dS] spacetimes, incorporating spontaneous Lorentz symmetry breaking (LSB) via an antisymmetric tensor field. Focusing on two metric configurations, we derive general analytical expressions for the horizon radius, photon sphere, shadow radius, and weak gravitational deflection angle. The universality of these expressions enables their applicability to a broad class of non-rotating spacetimes beyond KR gravity. By confronting these models with empirical data from the Event Horizon Telescope (EHT) and Solar System experiments, the work yields tight constraints on the Lorentz-violating parameter $\ell$, demonstrating that certain parameterizations - particularly Case B - yield observationally viable and physically consistent outcomes. Additionally, we employ topological methods to analyze black hole thermodynamics and photon sphere stability, uncovering critical geometrical structures in both thermodynamic and optical contexts. These multimodal signatures collectively offer a powerful framework for testing Lorentz-violating extensions of General Relativity (GR) and highlight the potential of topological diagnostics in gravitational physics.