Thermodynamic topology of Kiselev-AdS black holes within f (R, T) gravity

Abstract In this paper, we investigate the topological charge and conditions for the existence of the photon sphere in Kiselev-anti-de Sitter (AdS) black holes within gravity. Furthermore, we establish their topological classifications. We employ two different methods based on Duan’s topological current ϕ -mapping theory viz analysis of temperature and the generalized Helmholtz free energy methods to study the topological classes of our black hole. Considering this black hole, we discuss the critical and zero points (topological charges and topological numbers) for different parameters. Our findings reveal that the Kiselev parameter ω and gravity parameter γ influence the number of topological charges of black holes, providing novel insights into topological classifications. We observe that for given values of the free parameters, total topological charges ( ) exist for the T method and total topological numbers ( ) for the generalized Helmholtz free energy method. Our research findings elucidate that, in contrast to the scenario in which , in other cases, increasing γ increases the number of total topological charges for the black hole. Interestingly, for the phantom field ( ), we observe that decreasing γ increases the number of topological charges. Additionally, we study the results for the photon sphere. The studied models reveal that the simultaneous presence of γ and ω effectively expands the permissible range for γ . In other words, the model can exhibit black hole behavior over a larger domain. Additionally, we observe that with the stepwise reduction of ω , the region covered by singularity diminishes and becomes more restricted. However, an interesting point about all three ranges is the elimination of the forbidden region in this model. In other words, this model and the investigated areas appear to have no region in which both the ϕ and metric functions simultaneously lack solutions. Additionally, we fully check the curvatures singularities and energy conditions for the mentioned black hole.