Paintings: a planar approach to higher dimensions
type: Item
instance of: scholarly article
title: Paintings: a planar approach to higher dimensions
zbMATH Open document ID: 0638.57012
DOI: 10.1007/BF00149269
author: Sóstenes Lins
published in: Geometriae Dedicata
publication date: 1986-01-01T00:00:00Z
review text: An (n\(+1)\)-coloured graph is a finite and regular graph of degree \(n+1\) with an (n\(+1)\)-colouring of the edges such that at each vertex there is exactly one edge of each colour. PL closed n-manifolds can be encoded by certain such \((n+1)\)-coloured graphs called n-crystallizations. In this paper a related encoding by sets of n planar bipartite directed graphs, called n-paintings, is obtained. It is shown that a theorem of \textit{M. Ferri} and \textit{C. Gagliardi} [Pac. J. Math. 100, 85-103 (1982; Zbl 0517.57003)] giving an equivalence of crystallizations corresponding to homeomorphism of manifolds, can be stated in a rather simple way (restricted crystallization moves become deletion and insertion of edges in planar graphs). Properties of 3-paintings and minimum (i.e. least number of vertices encoding the same manifold) 3-crystallizations are obtained, and two conjectures stated.
Mathematics Subject Classification ID: 57Q99
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 4040858
zbMATH Keywords: 3-manifolds
zbMATH Keywords: PL-manifolds
zbMATH Keywords: \((n+1)\)-coloured graph
zbMATH Keywords: \((n+1)\)-colouring of the edges
zbMATH Keywords: n-crystallizations
zbMATH Keywords: planar bipartite directed graphs
zbMATH Keywords: n- paintings
zbMATH Keywords: crystallization moves
reviewed by: Bjarne Toft
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00149269
OpenAlex ID: W2033060851
description: scientific article; zbMATH DE number 4040858