  Recent observations indicate that many if not all galaxies host massive
central black holes. In this paper we explore the influence of black holes on
the lensing properties. We model the lens as an isothermal ellipsoid with a
finite core radius plus a central black hole. We show that the presence of the
black hole substantially changes the critical curves and caustics. If the black
hole mass is above a critical value, then it will completely suppress the
central images for all source positions. Realistic central black holes likely
have masses below this critical value. Even in such sub-critical cases, the
black hole can suppress the central image when the source is inside a zone of
influence, which depends on the core radius and black hole mass. In the
sub-critical cases, an additional image may be created by the black hole in
some regions, which for some radio lenses may be detectable with
high-resolution and large dynamic-range VLBI maps. The presence of central
black holes should also be taken into account when one constrains the core
radius from the lack of central images in gravitational lenses.
