Hyperbolic 2-fold Branched Coverings

Abstract

In the Kirby list is presented the following problem: describe the equivalence classes in the set of knots under the relation K$_{1}$ is equivalent to K$_{2}$ if their 2-fold cyclic branched coverings are homeomorphic 3-manifolds. In this paper we consider the basic case of hyperbolic manifold. In the fi{}rst part of this paper we want to present briefl{}y the results, yet available in some previous works, which solve this problem. In the second part we present examples of knots with the same 2-fold branched covering which show that the theorem, which describes the possible relations between two knots in the same equivalence class, is the best possible.