Representing 3-manifolds by triangulations of S3: a constructive approach

In a paper of I. V. Izmestʹev and M. Joswig [Adv. Geom. 3 (2003), no. 2, 191–225;], it was shown that any closed orientable 3-manifold M arises as a branched covering over S3 from some triangulation of S3. The proof of this result is based on the fact that any closed orientable 3-manifold M is a sim...

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Autores: Hilden, Hugh Michael, Montesinos Amilibia, José María, Tejada Jiménez, Débora María, Toro Villegas, Margarita María
Tipo de documento: artigo
Data de publicação:2005
País:España
Recursos:Universidad Complutense de Madrid (UCM)
Repositório:Docta Complutense
Idioma:inglês
OAI Identifier:oai:docta.ucm.es:20.500.14352/50760
Acesso em linha:https://hdl.handle.net/20.500.14352/50760
Access Level:Acceso aberto
Palavra-chave:515.16
3-manifolds
Topología
1210 Topología
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spelling Representing 3-manifolds by triangulations of S3: a constructive approachHilden, Hugh MichaelMontesinos Amilibia, José MaríaTejada Jiménez, Débora MaríaToro Villegas, Margarita María515.163-manifoldsTopología1210 TopologíaIn a paper of I. V. Izmestʹev and M. Joswig [Adv. Geom. 3 (2003), no. 2, 191–225;], it was shown that any closed orientable 3-manifold M arises as a branched covering over S3 from some triangulation of S3. The proof of this result is based on the fact that any closed orientable 3-manifold M is a simple 3-branched covering over S3 with a knot K as branched set [H. M. Hilden, Amer. J. Math. 98 (1976), no. 4, 989–997; J. M. Montesinos, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;]. In the paper under review the authors obtain the same result in a different way, which turns out to be constructive. More precisely, a triangulation Δ of the 3-sphere S3 defines uniquely a number m≤4, a subgraph Γ of Δ and a representation ω(Δ) of π1(S3∖Γ) into the symmetric group of m indices Σm. The aim of the paper is to prove that if (K,ω) is a knot or a link K in S3 together with a transitive representation ω:π1(S3∖K)→Σm, 2≤m≤3, then there is a constructive procedure to obtain a triangulation Δ of S3 such that ω(Δ)=ω. This new method involves a new representation of knots and links, called a butterfly representation.Soc. Colombiana Mat.Universidad Complutense de Madrid20052005-01-0120052005-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/50760reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/507602026-06-02T12:44:21Z
dc.title.none.fl_str_mv Representing 3-manifolds by triangulations of S3: a constructive approach
title Representing 3-manifolds by triangulations of S3: a constructive approach
spellingShingle Representing 3-manifolds by triangulations of S3: a constructive approach
Hilden, Hugh Michael
515.16
3-manifolds
Topología
1210 Topología
title_short Representing 3-manifolds by triangulations of S3: a constructive approach
title_full Representing 3-manifolds by triangulations of S3: a constructive approach
title_fullStr Representing 3-manifolds by triangulations of S3: a constructive approach
title_full_unstemmed Representing 3-manifolds by triangulations of S3: a constructive approach
title_sort Representing 3-manifolds by triangulations of S3: a constructive approach
dc.creator.none.fl_str_mv Hilden, Hugh Michael
Montesinos Amilibia, José María
Tejada Jiménez, Débora María
Toro Villegas, Margarita María
author Hilden, Hugh Michael
author_facet Hilden, Hugh Michael
Montesinos Amilibia, José María
Tejada Jiménez, Débora María
Toro Villegas, Margarita María
author_role author
author2 Montesinos Amilibia, José María
Tejada Jiménez, Débora María
Toro Villegas, Margarita María
author2_role author
author
author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 515.16
3-manifolds
Topología
1210 Topología
topic 515.16
3-manifolds
Topología
1210 Topología
description In a paper of I. V. Izmestʹev and M. Joswig [Adv. Geom. 3 (2003), no. 2, 191–225;], it was shown that any closed orientable 3-manifold M arises as a branched covering over S3 from some triangulation of S3. The proof of this result is based on the fact that any closed orientable 3-manifold M is a simple 3-branched covering over S3 with a knot K as branched set [H. M. Hilden, Amer. J. Math. 98 (1976), no. 4, 989–997; J. M. Montesinos, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;]. In the paper under review the authors obtain the same result in a different way, which turns out to be constructive. More precisely, a triangulation Δ of the 3-sphere S3 defines uniquely a number m≤4, a subgraph Γ of Δ and a representation ω(Δ) of π1(S3∖Γ) into the symmetric group of m indices Σm. The aim of the paper is to prove that if (K,ω) is a knot or a link K in S3 together with a transitive representation ω:π1(S3∖K)→Σm, 2≤m≤3, then there is a constructive procedure to obtain a triangulation Δ of S3 such that ω(Δ)=ω. This new method involves a new representation of knots and links, called a butterfly representation.
publishDate 2005
dc.date.none.fl_str_mv 2005
2005-01-01
2005
2005-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/50760
url https://hdl.handle.net/20.500.14352/50760
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Soc. Colombiana Mat.
publisher.none.fl_str_mv Soc. Colombiana Mat.
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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