Universal topological classes of black holes surrounded by quintessence

This work explores the universal topological classes of various black hole configurations immersed in a quintessence field. The results indicate that the Schwarzschild black hole surrounded by quintessence is of the $W^{1-}$ type, exhibiting thermodynamic instability in both extremal temperature limits. The introduction of rotation alters its topological nature to the $W^{0+}$ class, where the Kerr black hole displays coexistence of a stable small black hole and an unstable large black hole at low temperatures. In comparison, anti de Sitter (adS) black holes display different thermodynamic topologies. The Schwarzschild-adS solution corresponds to the $W^{0-}$ class, where an unstable small branch and a stable large branch emerge in the high temperatures. Meanwhile, the Kerr-adS classified as $W^{1+}$, maintains stability in both high- and low-temperature limits. Furthermore, the 4D/5D quintessential black hole belongs to the $W^{0+}$ class within the Einstein-Gauss-Bonnet framework. Overall, parameters such as rotation and Gauss-Bonnet in quintessence background field affect the topological classifications of the black holes, while the quintessence field primarily modifies the stability range without changing the topological class itself.