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Representing 3-manifolds by triangulations of S3: a constructive approach

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Hilden, Hugh Michael and Montesinos Amilibia, José María and Tejada Jiménez, Débora María and Toro Villegas, Margarita María (2005) Representing 3-manifolds by triangulations of S3: a constructive approach. Revista Colombiana de Matemáticas, 39 (2). pp. 63-86. ISSN 0034-7426

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Abstract

In a paper of I. V. Izmestʹev 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 Δ of the 3-sphere S3 defines uniquely a number m≤4, a subgraph Γ of Δ and a representation ω(Δ) of π1(S3∖Γ) into the symmetric group of m indices Σm. The aim of the paper is to prove that if (K,ω) is a knot or a link K in S3 together with a transitive representation ω:π1(S3∖K)→Σm, 2≤m≤3, then there is a constructive procedure to obtain a triangulation Δ of S3 such that ω(Δ)=ω. This new method involves a new representation of knots and links, called a butterfly representation.


Item Type:Article
Uncontrolled Keywords:3-manifolds
Subjects:Sciences > Mathematics > Topology
ID Code:22347
Deposited On:12 Jul 2013 12:55
Last Modified:23 Jan 2019 12:17

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