Software
One of the main features of crystallization theory relies on the purely combinatorial nature of the representing objects, which makes them particularly suitable for computer manipulation. This fact allows a computational approach to the study of PL n-manifolds, which has been performed by means of several functions, collected in a unified program, called DUKE III.
Other programs are available, which actually are independent from DUKE III but related to it: they realize suitable algorithmic procedures which either make use - as input – of the code of an edge-coloured graph (obtained by DUKE III) or supply - as output - the code of an edge-coloured graph which can be successively inserted in DUKE III in order to be analyzed and simplified, for recognition of the represented manifold...).
TORUS BUNDLE |
TORUS BUNDLE is based on an algorithmic construction of edge-coloured graphs representing 3-manifolds, which are torus bundles over S ^{1} (see [M.R.Casali, Representing and recognizing torus bundles over S ^{1} , Boletķn de la Sociedad Matemįtica Mexicana (special issue in honor of Fico), 10(3) (2005), 89-106]). The algorithm starts from a regular integer matrix A which describes the monodromy and contains at least one zero element. This program has allowed the topological recognition of all torus bundles among the 3-manifolds represented by the existing crystallization catalogues C ^{28} and ^{~} C ^{26.} Download TORUS BUNDLE (Windows .exe) program |
GM-COMPLEXITY |
GM-COMPLEXITY uses the algorithmic procedure described in [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology and its Applications 144 (1-3) (2004), 201-209], to estimate Matveev's complexity of a 3-manifold starting from the code of an associated edge-coloured graph (GM-complexity computation). This program has allowed to compute GM-complexity of all non-orientable 3-manifolds represented by edge-coloured graphs up to 26 vertices (catalogue
^{~}
C
^{26}
) and of all orientable 3-manifolds represented by edge-coloured graphs up to 28 vertices (catalogue
C
^{28}
); classes of manifolds for which the estimation is actually exact have been detected. Furthermore, a comparison between different notions of complexity has been performed: see [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology and its Applications 144 (1-3) (2004), 201-209] and [M.R. Casali - P.Cristofori, Computing Matveev's complexity via crystallization theory: the orientable case, Acta Applicandae Mathematicae (2006), 113-123]. Note that, in the first paper, the approach to the computation of Matveev's complexity via GM-complexity has allowed to complete the classification of non-orientable irreducible and
P
^{2}-irreducible 3-manifolds up to complexity 6. |
CATALOGUES |
CATALOGUES is a collection of algorithmic procedures, which can be used to construct essential catalogues of bipartite a nd/or non bipartite edge-coloured graphs representing all orientable and/or non-orientable n-manifolds triangulated by a given number of coloured n-simplices, and to classify (i.e. subdivide into homeomorphism classes) their elements, as a step toward the topological recognition of the involved manifolds. For details, see: crystallization catalogues. |