  In general relativity without a cosmological constant, a classical theorem
due to Hawking states that stationary black holes must be topologically
spherical. This result is one of the several ingredients that collectively
imply the uniqueness of the Kerr metric. If, however, general relativity
describes gravity inexactly at high energies or over cosmological scales,
Hawking's result may not apply, and black holes with non-trivial topology may
be, at least mathematically, permissible. While tests involving electromagnetic
and gravitational-wave data have been used to place tight constraints on
various theoretical departures from a Kerr description of astrophysical black
holes, relatively little attention has been paid to topological alternatives.
In this paper, we derive a new exact solution in an $f(R)$ theory of gravity
which admits topologically non-trivial black holes, and calculate observables
like fluorescent K$\alpha$ iron-line profiles and black hole images from
hypothetical astrophysical systems which house these objects, to provide a
theoretical basis for new tests of black hole nature. On the basis of
qualitative comparisons, we show that topologically non-trivial objects would
leave a strong imprint on electromagnetic observables and can be easily
distinguished from general-relativistic black holes in nearly all cases.
