The combinatorics of piecewise linear manifolds by colored graphs
The present paper deals with crystallization theory, which is an interesting combinatorial representation method for piecewise-linear (PL) manifolds of arbitrary dimension: see \textit{M. Ferri, C. Gagliardi} and \textit{L. Grasselli} [Aequationes Math. 31, 121--141 (1986; Zbl 0623.57012)] for a survey on the foundational aspects of the theory, \textit{M. R. Casali, P. Bandieri} and \textit{C. Gagliardi} [Atti Semin. Mat. Fis. Univ. Modena 49, Suppl., 283--337 (2001; Zbl 1420.57066)] and \textit{P. Bandieri} et al. [ibid. 58, 11--45 (2011; Zbl 1263.57018)] for subsequent improvements.NEWLINENEWLINEThe authors give an overview of classical results and techniques on crystallizations from a graph-theoretical point of view, paying particular attention to certain combinatorial invariants such as regular genus, complexity and average order. Several open problems and conjectures concerning the above invariants are also included in the present survey paper.NEWLINENEWLINEIt should be noted that, for some of the covered topics, the overview of the theory could be completed with significant improvements which are not reported by the authors: see for example \textit{M. R. Casali} and \textit{P. Cristofori} [J. Knot Theory Ramifications 17, No. 5, 579--599 (2008; Zbl 1163.57017)] (resp. \textit{M. R. Casali} [Acta Appl. Math. 54, No. 1, 75--97 (1998; Zbl 0913.57011)] and \textit{P. Bandieri, P. Cristofori} and \textit{C. Gagliardi} [J. Knot Theory Ramifications 18, No. 3, 381--395 (2009; Zbl 1177.57021)]) for the topological classification of orientable (resp. non-orientable) 3-manifolds up to reduced complexity 28, or \textit{M. R. Casali} [JP J. Geom. Topol. 10, No. 1, 41--62 (2010; Zbl 1250.57036)] for the complete identification of minimal 3-manifolds up to genus four. It is also worth noting the existence of known relationships among Matveev's complexity and some of the 3-dimensional invariants considered in the present paper see \textit{P. Bandieri} et al. [op. cit.].
scientific article; zbMATH DE number 6081597