Crystallization Catalogues and archives of PL manifolds with low gem-complexity

In virtue of the purely combinatorial nature of the representing objects, crystallization theory turns out to be particularly suitable to computer enumeration.

In particular, a collection of algorithmic procedures (called CATALOGUES) was created,  in order  to construct essential catalogues of bipartite and/or non bipartite edge-coloured graphs representing all orientable and/or non-orientable n-manifolds triangulated by a given number of coloured n-simplices, with n=3,4, and to classify (i.e. subdivide into homeomorphism classes) their elements, as a step toward the topological recognition of the involved manifolds.  

  • Up to now, the process of generation and analysis has been carried out, in dimension 3,  for all rigid crystallizations of 3-manifolds up to 30 vertices

The corresponding catalogues C30 and ~C30 are described in [M.R. Casali - P. Cristofori, Computing Matveev's complexity via crystallization theory: the orientable case, Acta Applicandae Mathematicae (2006), 113-123] and [M.R. Casali - P. Cristofori, A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra, Journal of Knot Theory and its Ramifications, 17 (5) (2008), 1-23] (see also [S.Lins, Gems, computers and attractors for 3-manifolds, Knots and Everything 5, World Scientific, 1995] for previous results up to 28 vertices) for the orientable case and in [M.R.Casali, Classification of non-orientable 3-manifolds admitting decompositions into 26 coloured tetrahedra, Acta Applicandae Math. 54 (1999), 75-77] and [P.Bandieri - P.Cristofori - C.Gagliardi Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra, J. Knot Theory Ramifications 18 (2009), 381-395] for the non-orientable case.

Details about existing 3-dimensional crystallization catalogues may be found in the page: "CATALOGUES OF 3-MANIFOLDS"

  • By means of the algorithmic procedures belonging to the collection CATALOGUES, tables of crystallizations have been obtained in dimension 4, too.

Note that, in dimension 4, the generation of manifolds catalogues implies the previous generation of all gems (not necessarily crystallizations) representing 3-dimensional spheres up to a fixed order; moreover, suitable sequences of combinatorial moves realizing the PL-classification of the represented 4-manifolds have to be chosen and implemented. For these reasons, in order to avoid the explosion of data, the program codes have been recently parallelized in order to obtain a better performance, within the Italian Supercomputing Resource Allocation (ISCRA) project “Cataloguing PL-manifolds in dimension 3 and 4 via crystallization theory", supported by CINECA.

The first results obtained in dimension 4 have been presented at the Workshop “TRIANGULATION”, organized by the Mathematical Research Institute of Oberwolfach (Germany):  see the extended abstract of the talk held on May 4th 2012 (published on Oberwolfach Reports: or the slides of the lecture.

The following table shows the initial output of the generating program:

By means of the classifying algorithm, efficiently implemented in dimension 4, the complete PL-classification of all closed orientable PL 4-manifolds up to gem-complexity 8 has been obtained.
Algorithms and classification theorems, together with related results and possible applications, are the subject of the following papers:

o   M.R.Casali, Catalogues of PL-manifolds and complexity estimations via crystallization theory, Oberwolfach Reports, Report No. 24/2012 - Workshop “TRIANGULATIONS” (April 29th - May 04th 2012), 58-61.   DOI:

o   A. Marani - M. Rivi - P. Cristofori, Generation of Catalogues of PL n-manifolds: Computational Aspects on HPC Systems, 14-th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Timisoara (2012).

o   M.R. Casali - P. Cristofori, Coloured graphs representing PL 4-manifolds, Electronic Notes in Discrete Mathematics 40 (2013), 83-87.   DOI:

o   M.R. Casali - P. Cristofori, Cataloguing PL 4-manifolds by gem-complexity, The Electronic Journal of Combinatorics 22 (4) (2015), #P4.25.

o   M.R. Casali – P. Cristofori – C. Gagliardi, Classifying PL 4-manifolds via crystallizations: results and open problems, in: "A Mathematical Tribute to Professor José María Montesinos Amilibia”, Universidad Complutense Madrid (2016). [ISBN: 978-84-608-1684-3].   Download here

o   B. Basak - M.R. Casali, Lower bounds for regular genus and gem-complexity of PL 4-manifolds, Forum Mathematicum 29 (4) (2017), 761-773. DOI:

o   M.R. Casali – P. Cristofori – C. Gagliardi, Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants, Rend. Istit. Mat. Univ. Trieste 52 (2020), 431-458. DOI:


Details about existing 4-dimensional crystallization catalogues may be found in the page: "CATALOGUES OF 4-MANIFOLDS"