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

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.