Nonorientable 3-manifolds admitting coloured triangulations with at most 30 tetrahedra

We present the census of all non-orientable, closed, connected 3-manifolds admitting a rigid crystallization with at most 30 vertices. In order to obtain the above result, we generate, manipulate and compare, by suitable computer procedures, all rigid non-bipartite crystallizations up to 30 vertices.Comment: 18 pages, 3 figure

Geometric group presentations: a combinatorial approach

In this paper we obtain combinatorial conditions for the geometricity of group presentations; our criterion holds both for orientable and for non-orientable manifolds

GEOMETRIC GROUP PRESENTATIONS: A COMBINATORIAL APPROACH

In this paper we obtain combinatorial conditions for the geometricity of group presentations; such a criterion holds both for orientable and for non-orientable manifolds.In this paper we obtain combinatorial conditions for the geometricityof group presentations; such a criterion holds both for orientable and fornon-orientable manifolds

A census of genus two 3-manifolds up to 42 coloured tetrahedra

We improve and extend to the non-orientable case a recent result of Karabas, Malicki and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.Comment: 24 pages, 3 figure

Complexity computation for compact 3-manifolds via crystallizations and Heegaard diagrams

The idea of computing Matveev complexity by using Heegaard decompositions has been recently developed by two different approaches: the first one for closed 3-manifolds via crystallization theory, yielding the notion of Gem-Matveev complexity; the other one for compact orientable 3-manifolds via generalized Heegaard diagrams, yielding the notion of modified Heegaard complexity. In this paper we extend to the non-orientable case the definition of modified Heegaard complexity and prove that for closed 3-manifolds Gem-Matveev complexity and modified Heegaard complexity coincide. Hence, they turn out to be useful different tools to compute the same upper bound for Matveev complexity.Comment: 12 pages; accepted for publication in Topology and Its Applications, volume containing Proceedings of Prague Toposym 201

Computing Matveev's complexity via crystallization theory: the boundary case

The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base $\mathbb D^2$ and two exceptional fibers and, therefore, for all torus knot complements.Comment: 27 pages, 14 figure

Computing Matveev's complexity via crystallization theory: the orientable case

By means of a slight modification of the notion of GM-complexity introduced in [Casali, M.R., Topol. Its Appl., 144: 201-209, 2004], the present paper performs a graph-theoretical approach to the computation of (Matveev's) complexity for closed orientable 3-manifolds. In particular, the existing crystallization catalogue C-28 available in [Lins, S., Knots and Everything 5, World Scientific, Singapore, 1995] is used to obtain upper bounds for the complexity of closed orientable 3-manifolds triangulated by at most 28 tetrahedra. The experimental results actually coincide with the exact values of complexity, for all but three elements. Moreover, in the case of at most 26 tetrahedra, the exact value of the complexity is shown to be always directly computable via crystallization theory

Presentazioni di gruppi e spine di 3-variet\ue0.

Si presentano recenti risultati sulla geometricit\ue0 di presentazioni di gruppi