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Butterflies and 3-manifolds. (Spanish: Mariposas y 3-variedades)

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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

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


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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
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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