Toroidal black holes in four dimensions

From a purely geometric (kinematic) perspective, black holes in four dimensional spacetimes can have event horizons with arbitrary topologies. It is only when energy conditions are imposed that the horizon's topology is constrained to be that of a sphere. Despite this, exploring exotic horizon topologies remains theoretically intriguing since it allows to unveil structural aspects of General Relativity and gain intuition on energy condition violations. In the axisymmetric case, besides the well-known spherical topology, only a toroidal topology is consistent with the symmetry. Complete solutions, describing the entire exterior region of such toroidal black holes without singularities, have not been reported yet. To the best of our knowledge, the construction we present here is the first explicit example of a toroidal black hole solution in four spacetime dimensions that is free of singularities in the external region.