How hyperbolic knots with homeomorphic cyclic branched coverings are related

We determine the exact geometric relation between two hyperbolic knots K and K′ such that the n-fold cyclic branched covering of K coincides with the m-fold cyclic branched covering of K′. If m and n are not powers of two the solution of the problem is known (complete solution for the case m=n, partial results for the case n different from m). In the present work, we give a complete solution to the problem for branching orders which are powers of two, and thus in particular also for the most basic case of 2-fold branched coverings.