This paper investigates the topological photon sphere from two perspectives: examining its existence and characteristics for different black hole models, and using it to classify the parameter space of black hole models. The authors explore Einstein-Yang-Mills non-minimal black holes, AdS black holes surrounded by Chaplygin-like dark fluid, and Bardeen-like black holes in Einstein-Gauss-Bonnet gravity. They propose a classification based on the existence and positioning of photon spheres, which helps determine parameter ranges that delineate whether a solution represents a black hole or a naked singularity. The paper illustrates the utility of the photon sphere as a test for establishing permissible and non-permissible parameter ranges within specific theories of black hole solutions. Key findings include the calculation of unstable photon spheres, the identification of regions where black holes or naked singularities exist, and the determination of forbidden regions where the model lacks a physical response. The authors also discuss the bidirectional relationship between parameters and the topological photon sphere, suggesting that this relationship can be used to control the behavior of black holes.This paper investigates the topological photon sphere from two perspectives: examining its existence and characteristics for different black hole models, and using it to classify the parameter space of black hole models. The authors explore Einstein-Yang-Mills non-minimal black holes, AdS black holes surrounded by Chaplygin-like dark fluid, and Bardeen-like black holes in Einstein-Gauss-Bonnet gravity. They propose a classification based on the existence and positioning of photon spheres, which helps determine parameter ranges that delineate whether a solution represents a black hole or a naked singularity. The paper illustrates the utility of the photon sphere as a test for establishing permissible and non-permissible parameter ranges within specific theories of black hole solutions. Key findings include the calculation of unstable photon spheres, the identification of regions where black holes or naked singularities exist, and the determination of forbidden regions where the model lacks a physical response. The authors also discuss the bidirectional relationship between parameters and the topological photon sphere, suggesting that this relationship can be used to control the behavior of black holes.