Decomposition of the vertex group of 3-manifolds
type: Item
instance of: scholarly article
title: Decomposition of the vertex group of 3-manifolds
zbMATH Open document ID: 0762.57009
DOI: 10.1016/0012-365X(92)90038-H
author: Sóstenes Lins
author: Louis Kauffman
published in: Discrete Mathematics
publication date: 1993-01-16T00:00:00Z
review text: A 4-graph \(G\) is a finite 4-regular graph with a proper 4-edge- coloration. For \(k\in\{0,1,2,3\}\) a \(k\)-residue in \(G\) is a connected component of a subgraph induced by all the edges of \(k\) pre-selected colors. A 3-gem (3-dimensional graph encoded manifold) is a 4-graph where the number of 0-residues (the vertices) plus the number of 3-residues equals the number of 2-residues. A 3-crystallization is a 3-gem \(G\) in which there are exactly 4 3-residues. Every closed 3-manifold can be represented by a crystallization [see \textit{M. Pezzana}, Atti Sem. Mat. Fis. Univ. Modena 23 (1974), 269-277 (1975; Zbl 0314.57005)]. Let \(G\) be a bipartite crystallization, thus representing a closed orientable 3- manifold \(M\). The colors of the edges are labelled by the numbers 0,1,2,3. The generators of the vertex group \(\xi(G)\) of \(G\) are a set of symbols in 1-1 correspondence with the vertex set of \(G\). Its relators are in 1-1 correspondence with the bicolored cycles of \(G\) (for details see the paper). The authors prove that the group \(\xi(M)\) (indeed \(\xi(G)\) is an invariant of \(M\)) is isomorphic to the free product of three copies of the fundamental group of \(M\) with \(\mathbb{Z}\). This result opens the possibility of searching for topologically invariant automorphisms of \(\xi\).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 20F34
zbMATH DE Number: 89517
zbMATH Keywords: 3-dimensional graph encoded manifold
zbMATH Keywords: 4-regular graph
zbMATH Keywords: 4-edge-coloration
zbMATH Keywords: \(k\)-residue
zbMATH Keywords: 3-crystallization
zbMATH Keywords: vertex group
zbMATH Keywords: fundamental group
zbMATH Keywords: topologically invariant automorphisms
reviewed by: Alberto Cavicchioli
MaRDI profile type: Publication
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cites work: Graph-encoded maps
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description: scientific article; zbMATH DE number 89517