0-Efficient Triangulations of 3-Manifolds

By W. Jaco, J. Hyam Rubinstein. 2002

178 citations recorded by the source corpus.

efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation orM can be shown to be one of the manifolds S 3 , RP 3 or L(3,1). Similarly, any triangulation of a com- pact, orientable, irreducible, @-irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal tr is de- fined. It is shown if M is a compact, orientable, irreducible, @-irreducible 3-manifold having no essential annuli and distinct from the 3-cell, then ◦ M admits an ideal triangulation; furthermore, it is shown that any ideal trian- gulation of such a 3-manifold can be modified to a 0-efficient ideal triangula- tion. A 0-efficient triangulation of a closed manifold has only one vertex or the manifold is S 3 and the triangulation has precisely two vertices. 0-efficient tri- angulations of 3-manifolds with boundary, and distinct from the 3-cell, have all their vertices in the boundary and then just one vertex in each bound- ary component. As tools, we introduce the concepts of barrier surface and shrinking, as well as the notion of crushing a triangulation along a normal sur- face. A number of applications are given, including an algorithm to construct an irreducible decomposition of a closed, orientable 3-manifold, an algorithm to construct a maximal collection of pairwise disjoint, normal 2-spheres in a closed 3-manifold, an alternate algorithm for the 3-sphere recognition prob- lem, results on edges of low valence in minimal triangulations of 3-manifolds, and a construction of irreducible knots in closed 3-manifolds.

Subfields: Computer Graphics and Computer-Aided Design, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5e5f21e3d64b5db430e41d44018cdb2b6cb81a07

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Citations in the corpus

Coverings and minimal triangulations of 3-manifolds (2009, methodology): “4 in [3] and Lemma 7 in [4] imply that M D S(3) , contradicting c.”
1-efficient triangulations and the index of a cusped hyperbolic 3-manifold (2013, methodology): “We can now use z T as a barrier (see [25]) to produce an embedded normal nonperipheral torus T , which is either essential and isotopic into a boundary cusp, or bounds a solid torus or cube with knotted hole.”
Layered-triangulations of 3-manifolds (2006, methodology): “An understanding of layered-triangulations of the solid torus is necessary in our work on 1–efficient triangulations [13]; however, we have carried it much further.”