CATALOGUES OF 3-MANIFOLDS
In dimension 3, all orientable (resp. non-orientable) closed connected 3-manifolds admitting a coloured triangulation with at most 2p tetrahedra can be easily identified by means of catalogue C^{2p} (resp. ^{~}C^{2p} ) of rigid bipartite (resp. non-bipartite) crystallizations up to 2p vertices.
Output data of the C++ program (described in [M.R. Casali, Classification of non-orientable 3-manifolds admitting decompositions into <= 26 coloured tetrahedra , Acta Appl. Math. 54 (1999), 75-97]) generating C^{2p} and/or ^{~}C^{2p} , for a fixed p , are presented in the following table, according to the number of vertices.
The topological identification of involved 3-manifolds, up to PL-homeomorphisms, has been performed in the following cases, by means of different techniques:
- Non-orientable case: catalogue ^{~}C^{26}
In [M.R.Casali, Classification of non-orientable 3-manifolds admitting decompositions into <= 26 coloured tetrahedra , Acta Appl. Math. 54 (1999), 75-97], elements of ^{~}C^{26} are partitioned into classes via combinatorial moves; then, each class is analysed and identified “by hand” starting from the fundamental group of the represented manifold.
The obtained results are collected in the following statement:
Exactly seven closed connected prime non-orientable 3-manifolds exist, with gem-complexity less or equal to 12 (i.e., which admit a coloured triangulation consisting of at most 26 tetrahedra); they are exactly the non-orientable S^{2} -bundle over S^{1} , the product between the real projective plane and S^{1} , the torus bundle with monodromy given by a suitable order two unimodular A and the four Euclidean non-orientable 3-manifolds.
- Orientable case: catalogue C^{28}
Catalogue C^{28} was originally obtained and analysed by Lins, in [S.Lins, Gems, computers and attractors for 3-manifolds , Knots and Everything 5, World Scientific, 1995], thus proving the existence of exactly sixty nine closed connected prime orientable 3-manifolds up to gem-complexity 13 (i.e., which admit a coloured triangulation consisting of at most 28 tetrahedra). However, Lins identifies seventeen of these manifolds simply by means of their (different) fundamental groups.
In [M.R. Casali, Representing and recognizing torus bundles over S^{1} , Boletin de la Sociedad Matematica Mexicana (special issue in honor of Fico), 10 (3) (2004), 89-106], five elements of C^{28} are proved to be torus bundles over S^{1} .
Finally, in [M.R. Casali – P. Cristofori, Computing Matveev's complexity via crystallization theory: the orientable case, Acta Applicandae Mathematicae 92 (2) (2006), 113-123], an unambiguous identification of all elements of catalogue C^{28} is given, in terms of JSJ decompositions and fibering structures.
The detailed results are contained in the following file: catalogue C^{28}
For a summary, see Table 1.
- Orientable case: catalogue C^{30}
Catalogue C^{30} is obtained and analysed in [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], where a more refined use of elementary combinatorial moves yields an automatic partition of the crystallization catalogue into equivalence classes, which are proved to be in one-to one correspondence with the homeomorphism classes of the represented manifolds.
“ Γ-class ” is the program which implements the above algorithm, with respect to a fixed (finite) set of admissible sequences of elementary combinatorial moves.
details about the Γ-class program
The detailed results are contained in the following file: catalogue C^{30}
For a summary, see Table 2 (and the related explanatory notes).
A comparative analysis of both complexity and geometric properties of the manifolds represented in each subset of C^{30} consisting of all crystallizations with the same number of vertices, may be found in Table 3.
- Reduced catalogues of cluster-less crystallizations
In the last paragraph of [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], an additional hypothesis on the representing crystallizations is introduced, so to yield a considerable reduction of the catalogues without loss of generality as far as the represented 3-manifolds are concerned.
The numbers of rigid cluster-less crystallizations involved in the new catalogues C’^{2p} (and in the corresponding catalogues ^{~}C’^{2p} for non-orientable 3-manifolds, too) are shown in the following table:
Table 4:
rigid cluster-less crystallizations up to 30 vertices