Hyperbolic 2-fold branched coverings of links and their quotients

Abstract

Many 3-manifolds can be represented as 2-fold branched coverings of links, but this representation is, in general, not unique. In the Seifert fibered case the problem is usually local: For example, if K is a Montesinos knot its 2-fold branched covering is Seifert fibered and there exists a complete system of local geometric modifications on K by which we can get every other Montesinos knot with the same 2-fold branched covering. On the other hand, if the 2-fold covering M of a knot is hyperbolic, the situation is globally determined by the structure of the isometry group of M. In this paper we develop a global approach for the case that M is hyperbolic and we study the orbifolds which are quotients of M by the action of a 2-group of isometries. This leads to a complete description of the geometry of the possible configurations of knots with the same 2-fold branched coverings. Moreover we are also able to settle the 2-component link case, which was still open, by finding an explicit bound on the number of in-equivalent 2-component links which have the same hyperbolic 2-fold branched coverings.