Topological Classes of BTZ Black Holes

In a recent paper, black holes were viewed as topological thermodynamic defects using generalized off-shell free energy. The aforementioned work indicates that all black hole solutions in the pure Einstein–Maxwell gravity theory could be classified into three different topological classes for four and higher spacetime dimensions. In this paper, we investigate the topological number of BTZ black holes in distinct theories with different charges (Q) and rotational parameters (J). Using generalized free energy and Duan’s ϕ-mapping topological current theory, we found only two topological classes for BTZ spacetime. Particularly, for a BTZ black hole with rotation or in the Einstein–Power–Maxwell theory, there is only one zero point and the total topological number is 1. While for a BTZ black hole in new massive gravity, the global topological charge depends on the value of the specific parameter m, which provides a counter-example for the conjecture that the topological number is independent of the black hole’s parameters.