Determining knots and links by cyclic branched coverings
type: Item
instance of: scholarly article
title: Determining knots and links by cyclic branched coverings
zbMATH Open document ID: 0896.57006
DOI: 10.1023/A:1004945308468
published in: Geometriae Dedicata
publication date: 1998-10-01T00:00:00Z
review text: A cyclic covering \(M\) of \(S^3\) branched along a link \(L\) is ``strongly cyclic'', if each meridian of \(L\) is projected onto a generator of the cyclic group. Let \(M\) be a hyperbolic 3-manifold and let the covering group act by isometries (\(L\) is then \(2\pi/n\)-hyperbolic). The main result states: If \(M\) and \(M'\) are two such manifolds and \(L\) and \(L'\) the respective links, then the homeomorphy of \(M\) and \(M'\) implies the equivalence of \(L\) and \(L'\), if \(n\geq 2\) does not divide the order of the orientation-preserving symmetry group \(\text{Sym}_+(L)\) of \(L\). The proof uses a simple argument concerning \(p\)-groups. The result is applied to maximally symmetric \(D_6\)-manifolds (\(D_6=\) dihedral group of order 12) which turn out to be the 3-fold cyclic coverings of 2-bridge links.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1038318
zbMATH Keywords: hyperbolic link
zbMATH Keywords: cyclic branched covering
zbMATH Keywords: hyperbolic 3-manifold
author: Bruno Zimmermann
reviewed by: Gerhard Burde
MaRDI profile type: Publication
full work available at URL: a:1004945308468
OpenAlex ID: W39416703
description: scientific article; zbMATH DE number 1038318