On \(\pi\)-hyperbolic knots with the same 2-fold branched coverings
type: Item
instance of: scholarly article
title: On \(\pi\)-hyperbolic knots with the same 2-fold branched coverings
zbMATH Open document ID: 0955.57006
author: Marco Reni
published in: Mathematische Annalen
publication date: 2001-02-04T00:00:00Z
review text: This paper studies to what extent a \(\pi\)-hyperbolic knot is determined by its \(2\)-fold branched covering. A knot \(K\) is called \(\pi\)-hyperbolic if there is a singular hyperbolic structure on the \(3\)-sphere with singular locus \(K\) and cone angle \(\pi\) at each singular point. A knot is known to be \(\pi\)-hyperbolic iff its \(2\)-fold branched covering is hyperbolic and the covering involution is an isometry. The main result of the paper states that there are at most nine conjugacy classes of nonfree involutions in the group of orientation-preserving isometries of a closed oriented hyperbolic \(\mathbb Z_2\)-homology \(3\)-sphere. In particular, there are at most nine \(\pi\)-hyperbolic knots with the same \(2\)-fold branched covering. According to the author, the largest known number of \(\pi\)-hyperbolic knots with the same \(2\)-fold branched covering is four. The proof depends on a characterization of finite Sylow \(2\)-subgroups of the orientation-preserving diffeomorphism group of a \(\mathbb Z_2\)-homology \(3\)-sphere.
reviewed by: Igor Belegradek
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1463209
zbMATH Keywords: hyperbolic knot
zbMATH Keywords: branched cover
MaRDI profile type: Publication
full work available at URL: s002080050349
OpenAlex ID: W2087075867
DOI: 10.1007/S002080050349
description: scientific article; zbMATH DE number 1463209