Fox coloured knots and triangulations of S3

dc.contributor.authorHilden, Hugh Michael
dc.contributor.authorMontesinos Amilibia, José María
dc.contributor.authorTejada Jiménez, Débora María
dc.contributor.authorToro Villegas, Margarita María
dc.description.abstractA 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.
dc.description.departmentDepto. de Álgebra, Geometría y Topología
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.sponsorshipthe Group of Mathematics UNALMedellín
dc.identifier.citationG. Burde and H. Zieschang. Knots (Walter de Gruyter, 1985). R. H. Crowell and R. H. Fox. Introduction to knot theory. Reprint of the 1963 original. Graduate Texts in Mathematics 57 (Springer-Verlag, 1977). A. Edmonds and C. Livingston. Symmetric representations of knot groups. Topology Appl. 18, no. 2–3 (1984), 281–312. J. Goodman and H. Onishi. Even triangulations of S3 and the coloring of graphs. Trans. Amer. Math. Soc. 246 (1978), 501–510. H. M. Hilden. Every closed orientable 3-manifold is a 3-fold branched covering space of S3. Bull. Amer. Math. Soc. 80 (1974), 1243–1244. H. M. Hilden, J. M. Montesions, D. M. Tejada and M. M. Toro. A new representation of links: Butterflies. Preprint (2005). H. M. Hilden, J. M. Montesions, D. M. Tejada and M. M. Toro. Representing 3-manifolds by triangulations of S3: a constructive approach. Rev. Colombiana Mat. 39 (2005), 63–86. H. M. Hilden, J. M. Montesions, D. M. Tejada and M. M. Toro. Mariposas y 3-variedades. Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 28 (2004), 71–78. I. Izmestiev and M. Joswig. Branched coverings, triangulations and 3-manifolds. Adv. Geom. 3, no. 2 (2003), 191–225. M. Joswig. Projectivities in simplicial complexes and colorings of simple polytopes. Topology 23 (1984), 195–209. R. Lickorish. An Introduction to Knot Theory. Graduate texts in Mathematics 175 (Springer-Verlag, 1997). J. M. Montesions. A representation of closed orientable 3-manifolds as 3-fold branched coverings of S3. Bull. Amer. Math. Soc. 80 (1974), 845–846. J. M. Montesions. Lectures on 3-fold simple coverings and 3-manifolds. Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982), 157–177. Contemp. Math. 44 (1985). L. P. Neuwirth. Knot groups. Ann. Math. Stud. 56 (1965). R. Piergallini. Four-manifolds as 4-fold branched covers of S4. Topology 34, no. 3 (1995), 497–508.
dc.journal.titleMathematical Proceedings of the Cambridge Philosophical Society
dc.publisherCambridge Univ Press
dc.relation.projectIDBMF-2002- 04137-C02-01.
dc.rights.accessRightsrestricted access
dc.subject.keywordtriangulations of S3.
dc.subject.unesco1210 Topología
dc.titleFox coloured knots and triangulations of S3
dc.typejournal article
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