Thermodynamic topology of AdS black holes within non-commutative geometry and Barrow entropy

In this paper, we explore the thermodynamic topology of Schwarzschild-AdS black holes within the framework of non-commutative geometry and Barrow entropy. We provide an overview of the Schwarzschild-AdS black hole and discuss the significance of Barrow entropy. We determine the topological charges and discuss the implications of topological classifications. Key findings include the impact of the strength of the noncommutative geometry parameter α and deformation parameter δ on the number of topological charges. When δ raised from Barrow entropy is zero, the structure faces Bekenstein-Hawking entropy, resulting in a single topological charge $(\omega =+1)$. However, the presence of parameter δ leads to variations in the number of topological charges. However, with specific values of α, δ the topological charges are determined as $(\omega =+1,-1)$. We also examine the photon spheres of the black hole for different values of the parameter α which showed the photon spheres with topological charge (−1), indicating the preservation of the black hole structure. These results are illustrated and summarized in various figures and tables.