Abstract

HILDEN, Mike; MONTESINOS, José M.; TEJADA, Débora  and  TORO, Margarita. Representing 3-manifolds by triangulations of S³: a constructive approach. Rev.colomb.mat. [online]. 2005, vol.39, n.2, pp.63-86. ISSN 0034-7426.

A triangulation Δ of S³ defines uniquely a number m ≤ 4; a subgraph T of Δ and a representation ω(Δ) of Π1(S³\T) into Σ_m: It is shown that every (K,ω), where K is a knot or link in S³ and ω is transitive representation of Π1(S³\K) in Σ_m, 2 ≤ m ≤ 3, equals ω(Δ), for some Δ. From this, a representation of closed, orientable 3-manifolds by triangulations of S³ is obtained. This is a theorem of Izmestiev and Joswig, but, in contrast with their proof, the methods in this paper are constructive. Some generalizations are given. The method involves a new representation of knots and links, which is called a butter y representation.

Keywords : Knot; Link; 3-manifold; Triangulation; Representation; Branched covering; Coloration.

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