Classifying topology of consistent thermodynamics of the four-dimensional neutral Lorentzian NUT-charged spacetimes

In this paper, via employing the uniformly modified form of the generalized off-shell Helmholtz free energy, we investigate the topological numbers for the four-dimensional neutral Lorentzian Taub-NUT, Taub-NUT-AdS and Kerr-NUT spacetimes, and find that these solutions can also be classified into one of three types of those well-known black hole solutions, which implies that these spacetimes should be viewed as generic black holes from the viewpoint of the thermodynamic topological approach.