How hyperbolic knots with homeomorphic cyclic branched coverings are related
The paper is devoted to the following basic problem: how are two knots \(K\) and \(K'\) related such that the \(n\)-fold cyclic branched covering of \(K\) and the \(m\)-fold cyclic branched covering of \(K'\) give the same hyperbolic 3-manifold \(M\) (assuming also that the covering groups act by isometries; this is implied, however, by the orbifold geometrization theorem as announced by Thurston). If \(m\) and \(n\) are not powers of two, a solution has been given by the reviewer for the case \(m=n\) [Math. Ann. 311, No. 4, 665-673 (1998; Zbl 0913.57008)], and a partial solution by \textit{M. Reni} and the reviewer for the case \(m\neq n\) [Comment. Math. Helv. 76, No. 2, 300-313 (2001; Zbl 0990.57003)]. For the case of 2-fold branched coverings, \textit{M. Reni} gave a ``global'' solution by showing that there are at most nine knots with the same hyperbolic 2-fold branched covering [Math. Ann. 316, No. 4, 681-697 (2000; Zbl 0955.57006)] (the difference between these cases lies in the fact that, in general, the Sylow 2-subgroup of the isometry group of \(M\) is more complicated than the Sylow \(p\)-subgroups for primes \(p>2\)). In the present paper, a complete solution is given for the case that \(m\) and \(n\) are powers of two, and thus in particular for the case of 2-fold branched coverings, by describing how any two knots with coinciding cyclic branched coverings \(M\) can be constructed. For example, if \(m\) and \(n\) are different from two, then there are at most two such knots which are related by a ``standard abelian construction'': starting with a link \(L\) of two trivial components, the preimages of each of the two components in a cyclic branched covering along the other component give two knots \(K\) and \(K'\) in the 3-sphere with the same manifold \(M\) as a cyclic branched covering (which is the branched regular (\(\mathbb Z_m \times \mathbb Z_n\))-covering of the link \(L\)). If one or both of \(m\) and \(n\) are equal to two, the situation is more complicated and there are various other ways how to construct the knots \(K\) and \(K'\). (Together with the above mentioned papers, the basic case which remains open is that of prime numbers \(m \neq n\) different from two, for a hyperbolic 3-manifold \(M\) with non-solvable isometry group).
scientific article; zbMATH DE number 1787541
scientific article