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Resumen:
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A Fox coloured link is a pair (L,?), where L is a link in S3 and ? a simple and transitive representation of ?1(S3?L) onto the symmetric group ?3 on three elements. Here, a representation is called simple if it sends the meridians to transpositions. By works of the first two authors, any Fox coloured link (L,?) gives rise to a closed orientable 3-manifold M(L,?) equipped with a 3-fold simple covering p:M(L,?)?S3 branched over L, and any closed orientable 3-manifold is homeomorphic to an M(K,?) for some Fox coloured knot (K,?) [see H. M. Hilden, Bull. Amer. Math. Soc. 80 (1974), 1243–1244; J. M. Montesinos, Bull. Amer. Math. Soc. 80 (1974), 845–846;]. In [Adv. Geom. 3 (2003), no. 2, 191–225;], I. V. Izmest?ev and M. Joswig proved that a triangulation of S3 gives rise in a natural way to some graph G on S3 and a representation of ?1(S3?G) into the symmetric group ?m for some m?4. They also proved that any pair (L,?), where L is a link in S3 and ? a simple (not necessarily transitive) representation of ?1(S3?L) into the symmetric group ?4, can be obtained from a triangulation of S3. The proof that Izmest?ev and Joswig gave of this result is non-constructive. In the paper under review, the authors give a constructive proof of the same result. In particular, given a pair (L,?) consisting of a link L in S3 and a simple (not necessarily transitive) representation of ?1(S3?L) onto the symmetric group ?4, they construct a triangulation of S3 that gives rise to (L,?) in a natural way.
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