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Hilden, Hugh Michael and Montesinos Amilibia, José María and Tejada Jiménez, Débora María and Toro Villegas, Margarita María
(2005)
*Representing 3-manifolds by triangulations of S3: a constructive approach.*
Revista Colombiana de Matemáticas, 39
(2).
pp. 63-86.
ISSN 0034-7426

PDF
339kB |

Official URL: http://www.scm.org.co/index.php?option=com_wrapper&view=wrapper&Itemid=176

## Abstract

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.

Item Type: | Article |
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Uncontrolled Keywords: | 3-manifolds |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 22347 |

Deposited On: | 12 Jul 2013 12:55 |

Last Modified: | 23 Jan 2019 12:17 |

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