Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer \( n \ge 2 \). In the present paper, we study the following more general situation. Given two integers m and n, how are knots K1 and K2 related such that the m-fold cyclic branched covering of K1 coincides with the n-fold cyclic branched covering of K2. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S3. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds)