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Fox coloured knots and triangulations of S3

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Hilden, Hugh Michael and Montesinos Amilibia, José María and Tejada Jiménez, Débora María and Toro Villegas, Margarita María (2006) Fox coloured knots and triangulations of S3. Mathematical Proceedings of the Cambridge Philosophical Society, 141 (3). pp. 443-463. ISSN 0305-0041

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Abstract

A Fox coloured link is a pair (L,ω), where L is a link in S3 and ω a simple and transitive representation of π1(S3∖L) onto the symmetric group Σ3 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,ω) gives rise to a closed orientable 3-manifold M(L,ω) equipped with a 3-fold simple covering p:M(L,ω)→S3 branched over L, and any closed orientable 3-manifold is homeomorphic to an M(K,ω) for some Fox coloured knot (K,ω) [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. Izmestʹev 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 π1(S3∖G) into the symmetric group Σm for some m≤4. They also proved that any pair (L,ω), where L is a link in S3 and ω a simple (not necessarily transitive) representation of π1(S3∖L) into the symmetric group Σ4, can be obtained from a triangulation of S3. The proof that Izmestʹev 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,ω) consisting of a link L in S3 and a simple (not necessarily transitive) representation of π1(S3∖L) onto the symmetric group Σ4, they construct a triangulation of S3 that gives rise to (L,ω) in a natural way.


Item Type:Article
Uncontrolled Keywords:knots; triangulations of S3.
Subjects:Sciences > Mathematics > Topology
ID Code:22350
Deposited On:12 Jul 2013 12:58
Last Modified:12 Dec 2018 15:13

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