A note about complexity of Lens spaces
type: Item
instance of: scholarly article
title: A note about complexity of Lens spaces
zbMATH Open document ID: 1339.57018
author: Maria Rita Casali
author: Paola Cristofori
published in: Forum Mathematicum
publication date: 2015-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.5728
review text: For a closed \(3\)-manifold \(M\) its complexity \(c(M)\), as defined by Matveev, is the minimum number of true vertices among all almost simple spines of \(M\). For a lens space \(L(p,q)\) it turns out that \(c(L(p,q))\leq S(p,q)-3\), where \(S(p,q)\) is the sum of all partial quotients in the expansion of \(p/q\) as a regular continued fraction. Jaco, Rubinstein and Tillmann [\textit{W. Jaco} et al., Algebr. Geom. Topol. 11, No. 3, 1257--1265 (2011; Zbl 1229.57010)] showed equality for some infinite families of lens spaces. In the present paper the authors relate \(c(M)\) to the \textit{gem}-complexity \(k(M)\) and the \textit{Gem-Matveev}-complexity \(c_{GM}(M)\), which are defined in terms of crystallizations of \(M\). (A crystallization of a closed triangulated \(3\)-manifold \(M\) is a representation of \(M\) by a certain \(4\)-colored graph). The main result is that \(c_{GM}(L(p,q))\leq S(p,q)-3\), for all \(p\geq 3\), and \(k(L(p,q))\leq 2S(p,q)-1\), for all \(p\geq 2\). Furthermore, using results arising from the crystallization catalogues, the authors show that there are exactly \(110\) closed prime orientable \(3\)-manifolds \(M\) with \(k(M)=5\), and they analyze their geometries and the relation between \(k(M)\) and \(c(M)\).
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M15
zbMATH DE Number: 6505798
zbMATH Keywords: complexity of \(3\)-manifolds
zbMATH Keywords: lens spaces
reviewed by: Wolfgang Heil
MaRDI profile type: Publication
OpenAlex ID: W2012825852
arXiv ID: 1309.5728
DOI: 10.1515/FORUM-2013-0185
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description: scientific article