We give a nearly complete solution of the problem of how many different knots and links in the 3-sphere, and more generally in homology 3-spheres, can have the same hyperbolic 3-manifold as their common 2-fold branched covering. This number depends on the number of components of the links. We show that the best possible upper bound is 9 for knots and for 2-component links, and 3 for links with more than two components.
