A tongue-in-cheek answer on Worldbuilding got me wondering: what happens to the shape of a black hole when other large masses happen near? The idea of distorting it is not completely out of the question since a rotating black hole will become oblate and develop an ergosphere.
A later, more serious question concerns the shape of the horizon with two black holes near each other.
If you take a simple model of a potential field, the isosurface of a given escape velocity will show tidal bulges. But with the connection between black hole surface area and entropy, I wonder how it can be changed without affecting some innate property of the BH, which an external object is not.