How hyperbolic knots with homeomorphic cyclic branched coverings are related
type: Item
instance of: scholarly article
title: How hyperbolic knots with homeomorphic cyclic branched coverings are related
zbMATH Open document ID: 0997.57004
DOI: 10.1016/S0166-8641(01)00091-8
author: Mattia Mecchia
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: 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 mor…
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1787541
zbMATH Keywords: cyclic branched covering
zbMATH Keywords: hyperbolic knot
zbMATH Keywords: hyperbolic 3-manifold
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
cites work: Q5658847
cites work: Three-dimensional orbifolds and cone-manifolds
cites work: Q3973938
cites work: Knots with Free Period
cites work: Homology Invariants
cites work: Hyperbolic 3-manifolds singular along knots
cites work: Q3733804
cites work: Hyperbolic 2-fold branched coverings of links and their quotients.
cites work: On \(\pi\)-hyperbolic knots with the same 2-fold branched coverings
cites work: Hyperbolic 3-manifolds as cyclic branched coverings
cites work: Q3926778
cites work: On involutions of the 3-sphere
cites work: On hyperbolic knots with homeomorphic cyclic branched coverings
Recommended article: Hyperbolic 2-fold branched coverings of links and their quotients.
Recommended article: Q4799224
Recommended article: On \(\pi\)-hyperbolic knots and branched coverings
Recommended article: Standard situations for cyclic branched coverings of hyperbolic knots
Recommended article: On the Isometry Groups of Cyclic Branched Coverings of Hyperbolic Knots
Recommended article: On hyperbolic knots with the same \(m\)-fold and \(n\)-fold cyclic branched coverings
Recommended article: On hyperbolic knots with homeomorphic cyclic branched coverings
Recommended article: The number of knots and links with the same 2-fold branched covering
Recommended article: THREE CYCLIC BRANCHED COVERS SUFFICE TO DETERMINE HYPERBOLIC KNOTS
Recommended article: Conway irreducible hyperbolic knots with two common covers
description: scientific article; zbMATH DE number 1787541