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 C2p (resp. ~C2p ) 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 C2p and/or ~C2p , 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 ~C26
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 ~C26 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 S2 -bundle over S1 , the product between the real projective plane and S1 , the torus bundle with monodromy given by a suitable order two unimodular A and the four Euclidean non-orientable 3-manifolds.
- Orientable case: catalogue C28
Catalogue C28 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 S1 , Boletin de la Sociedad Matematica Mexicana (special issue in honor of Fico), 10 (3) (2004), 89-106], five elements of C28 are proved to be torus bundles over S1 .
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 C28 is given, in terms of JSJ decompositions and fibering structures.
The detailed results are contained in the following file: catalogue C28
For a summary, see Table 1.
- Orientable case: catalogue C30
Catalogue C30 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 C30
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 C30 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