Special representations for \(n\)-bridge links
type: Item
instance of: scholarly article
title: Special representations for \(n\)-bridge links
zbMATH Open document ID: 0826.57006
DOI: 10.1007/BF02574362
published in: Discrete \& Computational Geometry
publication date: 1994-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/131314
review text: The Schubert normal form of a 2-bridge link, \(K(p, q)\), consists of two overpasses (called `bridges') and two underpasses, and is characterized by two coprime integers \(p\), \(q\) \((1\leq p<q)\). Here \(p-1\) is the number of the undercrossings of each of the bridges, and \(q\) indicates the undercrossings to which the endpoints of the bridges are connected. Schubert classified all 2-bridge links in terms of this normal form. Generalizing this presentation, the authors show that any 3-bridge link can be represented by a special type of diagram that is characterized by six integers. They show that this presentation is not unique using some results of \textit{O. Morikawa} [Math. Semin. Notes, Kobe Univ. 9, 349-369 (1981; Zbl 0486.57005), Yokohama Math. J. 30, 53-72 (1982; Zbl 0526.57004)], who studied a class of 3-bridge links. Also, they have no general criterion for classifying 3-bridge links. It is known [\textit{J. S. Birman} and \textit{H. M. Hilden}, Trans. Am. Math. Soc. 213, 315-352 (1975; Zbl 0312.55004)] that any closed orientable 3- manifold with Heegaard genus two is a two-fold branched cover of \(S^3\) branched over a 3-bridge link. Using this presentation, the authors represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, they prove some results about these 3-manifolds and extend the 3-bridge presentation to \(n\)-bridge links.
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 05C15
zbMATH DE Number: 617321
zbMATH Keywords: manifold crystallizations
zbMATH Keywords: edge colored graphs
zbMATH Keywords: Schubert normal form of a 2-bridge link
zbMATH Keywords: 3-bridge link
zbMATH Keywords: genus two 3-manifolds
zbMATH Keywords: \(n\)-bridge links
author: Alberto Cavicchioli
author: B. Ruini
reviewed by: Taizo Kanenobu
MaRDI profile type: Publication
cites work: Heegaard Splittings of Branched Coverings of S 3
cites work: Knots
cites work: Representing branched coverings by edge-coloured graphs
cites work: Neuwirth manifolds and colourings of graphs
cites work: Crystallisations of 2-Fold Branched Coverings of S 3
cites work: A graph-theoretical representation of PL-manifolds -- a survey on crystallizations
cites work: Q3907519
cites work: Q4114528
cites work: Three-Fold Branched Coverings of S 3
cites work: Q3266916
cites work: On Knots. (AM-115)
cites work: Q4061696
cites work: Branched coverings
cites work: THREE-MANIFOLDS AS 3-FOLD BRANCHED COVERS OF S3
cites work: Q3946819
cites work: Q3040142
cites work: Seifert manifolds
cites work: Simplifying spines of 3-manifolds
cites work: Q4074117
cites work: Q4107635
cites work: Q5667781
cites work: Knoten mit zwei Brücken
cites work: Q3687352
Recommended article: Q4027125
Recommended article: Representations of a link group
Recommended article: \(SU(2)\)-representation spaces for two-bridge knots
Recommended article: New formulas for the character varieties of two-bridge links
Recommended article: NONABELIAN REPRESENTATIONS OF 2-BRIDGE KNOT GROUPS
Recommended article: A combinatorial representation of links by quasitoric braids
Recommended article: Three-dimensional rotational averages in radiation-molecule interactions: an irreducible cartesian tensor formulation
Recommended article: Representation varieties of twisted Hopf links
Recommended article: Link polynomials related to the new braid group representations
Recommended article: Representations of (1,1)-knots
description: scientific article; zbMATH DE number 617321