On a class of hyperbolic 3-orbifolds of small volume and small Heegaard genus associated to 2-bridge links
type: Item
instance of: scholarly article
title: On a class of hyperbolic 3-orbifolds of small volume and small Heegaard genus associated to 2-bridge links
zbMATH Open document ID: 0947.57003
DOI: 10.1007/BF02904219
author: Mattia Mecchia
author: Bruno Zimmermann
publication date: 2000-11-01T00:00:00Z
review text: The authors study a certain class of \(3\)-orbifolds \(O_n(\alpha/\beta)\). The underlying space of these orbifolds is the \(3\)-sphere, and the singular set is a certain link with two specific edges added, and with specific branching orders. The link is the \(2\)-bridge link \(L(\alpha/\beta)\) (the twists are encoded in any expansion of the rational number \(\alpha/\beta\) as continued fraction), and the additional edges are the two standard (upper and lower) unknotting tunnels. The authors prove by an explicit construction that \(O_n(\alpha/\beta)\) is a quotient of the orbifold \(L_n(\alpha/\beta)\) by an action of the Klein group \(Z/2\times Z/2\), and the projection is a \(4\)-fold orbifold covering. If \(L_n(\alpha/\beta)\) is hyperbolic, the covering transformations can be chosen to be isometries, in particular \(O_n(\alpha/\beta)\) has also a hyperbolic structure and the volume of \(L_n(\alpha/\beta)\) is four times the one of \(O_n(\alpha/\beta)\). Here, \(L_n(\alpha/\beta)\) is an orbifold with underlying space the \(3\)-sphere and singular set the link \(L(\alpha/\beta)\) (with branching order \(n\)). The advantage is that for such orbifolds effective algorithms exist to decide whether they are hyperbolic and to compute their volume. The authors use this (more specifically, the program snappea), to compute the volumes for a long list of examples. They obtain some of the smallest known volumes of hyperbolic \(3\)-orbifolds.
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1462205
zbMATH Keywords: hyperbolic \(3\)-orbifold
zbMATH Keywords: \(2\)-bridge link
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02904219
OpenAlex ID: W1999499359
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published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 1462205