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Hilden, Hugh Michael and Montesinos Amilibia, José María and Tejada Jiménez, Débora María and Toro Villegas, Margarita María
(2004)
*Butterflies and 3-manifolds. (Spanish: Mariposas y 3-variedades).*
Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales., 28
(106).
pp. 71-78.
ISSN 0370-3908

PDF
Restringido a Repository staff only 314kB |

Official URL: http://www.accefyn.org.co/revista/Vol_28/106/71-78.pdf

## Abstract

A butterfly is a 3-ball B with an even number of polygonal faces, named wings, pair-wise identified. Each identification between two wings is required to be a topological reflexion whose axis is an edge shared by the wings. The set of axes of the identifications is called the thorax of the butterfly.

A knot K⊂S3 admits a butterfly representation if there is a butterfly B with thorax T such that, after the identifications, (B,T) is homeomorphic to (S3,K).

In this paper it is shown that any 3-colorable knot admits a butterfly representation (B,T) such that the butterfly B has a 4-colored triangulation compatible with the 3-coloration of the knot. By a result of H. M. Hilden [Amer. J. Math. 98 (1976), no. 4, 989–997;] and J. M. Montesinos [Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;], one can associate to any 3-manifold a 3-colored knot. A corollary of the main result of the paper is therefore that one can associate to any 3-manifold at least one butterfly.

Item Type: | Article |
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Uncontrolled Keywords: | Knots, Fundamental group, 3-manifolds, Branched coverings. |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 22315 |

Deposited On: | 11 Jul 2013 15:41 |

Last Modified: | 12 Dec 2018 15:13 |

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