  In this paper, we extend our previous work [Phys. Rev. D 107, 024024 (2023)]
to the more general cases with a negative cosmological constant, and
investigate the topological numbers for the singly rotating Kerr-AdS black
holes in all dimensions and the four-dimensional Kerr-Newman-AdS black hole as
well as the three-dimensional Ba\~nados-Teitelboim-Zanelli black hole. We find
that the topological numbers of black holes are remarkably influenced by the
cosmological constant. In addition, we also demonstrate that the dimension of
spacetimes has an important effect on the topological number for rotating AdS
black holes. Furthermore, it is interesting to observe that the difference
between the topological number of the AdS black hole and that of its
corresponding asymptotically flat black hole is always unity. This new
observation leads us to conjure that it might be valid also for other black
holes. Of course, this novel conjecture needs to be further verified by
examining the topological numbers of many other black holes and their AdS
counterparts in the future work.
