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Resumen:
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In a paper of I. V. Izmest?ev 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 ? of the 3-sphere S3 defines uniquely a number m?4, a subgraph ? of ? and a representation ?(?) of ?1(S3??) into the symmetric group of m indices ?m. The aim of the paper is to prove that if (K,?) is a knot or a link K in S3 together with a transitive representation ?:?1(S3?K)??m, 2?m?3, then there is a constructive procedure to obtain a triangulation ? of S3 such that ?(?)=?. This new method involves a new representation of knots and links, called a butterfly representation.
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