is there an equation/argument which describes the shape or defining properties of a black hole other than those of its event horizon and point of singularity?

also, if we describe a black hole as an entity which swallows light, bends space and is an object which has a "lip" over which matter falls into... then why are black holes sometimes shown with streams of energy/radiation coming out of the top and what seems to be a "bottom?"

thank you.
simon