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
inequivalent 2-component links which have the same hyperbolic 2-fold branched
coverings.
