  Given a link in a 3-manifold such that the complement is hyperbolic, we
provide two modifications to the link, called the chain move and the switch
move, that preserve hyperbolicity of the complement, with only a relatively
small number of manifold-link pair exceptions, which are also classified. These
modifications provide a substantial increase in the number of known hyperbolic
links in the 3-sphere and other 3-manifolds.
