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 ^~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

down-load 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