589 research outputs found
Inflations of ideal triangulations
Starting with an ideal triangulation of the interior of a compact 3-manifold
M with boundary, no component of which is a 2-sphere, we provide a
construction, called an inflation of the ideal triangulation, to obtain a
strongly related triangulations of M itself. Besides a step-by-step algorithm
for such a construction, we provide examples of an inflation of the
two-tetrahedra ideal triangulation of the complement of the figure-eight knot
in the 3-sphere, giving a minimal triangulation, having ten tetrahedra, of the
figure-eight knot exterior. As another example, we provide an inflation of the
one-tetrahedron Gieseking manifold giving a minimal triangulation, having seven
tetrahedra, of a nonorientable compact 3-manifold with Klein bottle boundary.
Several applications of inflations are discussed.Comment: 48 pages, 45 figure
Quadrilateral-octagon coordinates for almost normal surfaces
Normal and almost normal surfaces are essential tools for algorithmic
3-manifold topology, but to use them requires exponentially slow enumeration
algorithms in a high-dimensional vector space. The quadrilateral coordinates of
Tollefson alleviate this problem considerably for normal surfaces, by reducing
the dimension of this vector space from 7n to 3n (where n is the complexity of
the underlying triangulation). Here we develop an analogous theory for
octagonal almost normal surfaces, using quadrilateral and octagon coordinates
to reduce this dimension from 10n to 6n. As an application, we show that
quadrilateral-octagon coordinates can be used exclusively in the streamlined
3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing
experimental running times by factors of thousands. We also introduce joint
coordinates, a system with only 3n dimensions for octagonal almost normal
surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using
cohomology, plus other minor changes; v3: Minor housekeepin
Curvature bounds for surfaces in hyperbolic 3-manifolds
We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds
and use this to prove existence of universal bounds on the principal curvatures
of surfaces embedded in hyperbolic 3-manifolds.Comment: 21 pages, 9 figures, published version, added figures, fixed typo
Coverings and minimal triangulations of 3-manifolds
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space L(4k; 2k-1) and the generalised quaternionic space S/Q have complexity k, where k≥2. Moreover, it is shown that their minimal triangulations are unique
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