Lensing by a singular isothermal sphere and a black hole
  Most galaxies host central supermassive black holes. As two galaxies merge,
the black holes also merge. The final single black hole may suffer a kick due
to asymmetric gravitational radiation and may not be at the centre of the
galaxy; off-centre black holes may also be produced by other means such as
sustained acceleration due to asymmetric jet power. We model the main galaxy as
a singular isothermal sphere and the black hole as an off-centre point lens,
and study the critical curves and caustics using complex notation. We identify
the critical parameters that govern the transitions in the topology of critical
curves, caustics and pseudo-caustics, and find the number of images can be two,
three, four and five. We show examples of image configurations, including cases
where three highly de-magnified images are found close to the centre. The
perturbation on the image magnification due to the black hole scales linearly
with its mass in the off-centre case, and quadratically when the black hole is
at the centre. Such images are difficult to observe unless high-contrast and
high-resolution imaging facilities (e.g., the Square Kilometer Array in the
radio) become available.
