A note about complexity of Lens spaces
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 \(c(M)\).
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