Coverings and minimal triangulations of 3–manifolds

Jaco, William; Rubinstein, J. Hyam; Tillmann, Stephan

doi:10.2140/agt.2011.11.1257

Cited by 38 publications

(37 citation statements)

References 6 publications

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“…This is essentially similar to a construction for closed manifolds that appears in the function standard torus form() in [27, close cusps.c]. This layering construction is also analyzed in great detail by Jaco and Rubinstein [17]. We next go on to find a geometric realization of D , using the ideas of Section 1.…”

Section: Farey Combinatorics In Solid Torimentioning

confidence: 90%

“…As set out in Section 2, the combinatorics of the triangulation T are identical to a procedure found in the SnapPea kernel [27], called the layering construction by Jaco and Rubinstein [17]. Each integer˛near the middle of the continued fraction expansion gives rise to˛adjacent tetrahedra, to one edge of degree 2˛C 4, and to˛ 1 edges of degree 4 (the average degree of edges is always 6 by an Euler characteristic argument).…”

Section: Introductionmentioning

confidence: 99%

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Canonical triangulations of Dehn fillings

Guéritaud

Schleimer

2010

Geom. Topol.

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Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold (a similar result has been independently announced by Akiyoshi [4]). We also find the canonical decompositions of all hyperbolic Dehn fillings on one cusp of the Whitehead link complement. 51H20; 57M50 IntroductionLet M be a complete cusped hyperbolic 3-manifold of finite volume, and endow the cusps c 1 ; : : : ; c k of M with disjoint simple horoball neighborhoods H 1 ; : : : ; H k . The Ford-Voronoi domain F M consists of all points of M having a unique shortest path to the union of the H i . The complement of F is a compact complex C of totally geodesic polygons. By definition, the canonical decomposition D of M with respect to the H i has one 3-dimensional cell (an ideal polyhedron) per vertex of C , one face per edge of C , and one edge per (polygonal) face of C ; we say that D is dual to C . Other names for D are the geometrically canonical decomposition, or Delaunay (or Delone) decomposition. In [11], Epstein and Penner give a precise description of D in terms of convex hulls in Minkowski space R 3C1 . Weeks' program SnapPea [27] will compute D for most manifolds of moderate size.Akiyoshi [3] proves that, as the volumes of fH i g 1ÄiÄk vary, only finitely many decompositions D arise. By Mostow-Prasad rigidity, the resulting collection of Delaunay decompositions is a complete topological invariant of M . When M has a single cusp there is a unique Delaunay decomposition. When M has multiple cusps one may take all of their volumes to be equal; SnapPea uses the resulting decomposition for rigorous computation of isometry groups and detection of isometric manifolds (see Weeks [28] The present paper offers a relative result: we are interested in how the canonical decomposition D changes when the last cusp c k (where k 2) undergoes a Dehn filling along the slope s . Recall the operation of filling along s removes the interior of H k from M and glues a solid torus X s to the resulting boundary component, yielding the filled manifold M s . Thurston showed that the metric on M s Gromovconverges, with appropriate choices of basepoints, to the metric on M as the length of the filling slope s goes to infinity. Consequently, a Margulis tube (region where the injectivity radius is less than the Margulis constant) appears about the core curve ofExperimentation with SnapPea suggests that, for many manifolds, cusps, and slopes, after filling c k the polyhedra of D outside the Margulis tube undergo only a small geometric perturbation while the combinatorics of D s inside the tube has a predictable structure. To ensure such good behavior we choose the reference horoball neighborhoods fH i g 1Äi

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Section: Farey Combinatorics In Solid Torimentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Canonical triangulations of Dehn fillings

Guéritaud

Schleimer

2010

Geom. Topol.

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“…Twisted layered loops are conjectured by Matveev to have minimal complexity [26], and a proof of this claim has recently been announced by Jaco, Rubinstein and Tillmann [21]. Here we include four smaller cases (9 ≤ n ≤ 18) for standard coordinates, and four larger cases (30 ≤ n ≤ 75) for quadrilateral coordinates.…”

Section: Experimentationmentioning

confidence: 98%

Optimizing the double description method for normal surface enumeration

Burton

2010

Math. Comp.

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Abstract. Many key algorithms in 3-manifold topology involve the enumeration of normal surfaces, which is based upon the double description method for finding the vertices of a convex polytope. Typically we are only interested in a small subset of these vertices, thus opening the way for substantial optimization. Here we give an account of the vertex enumeration problem as it applies to normal surfaces and present new optimizations that yield strong improvements in both running time and memory consumption. The resulting algorithms are tested using the freely available software package Regina.

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“…Recall that the complexity of a cusped hyperbolic 3-manifold is defined as the number of tetrahedra in its minimal ideal triangulation. The exact values of complexity have been found for several infinite families of manifolds from the following classes: lens spaces and their coverings [6,7], hyperbolic manifolds with geodesic boundary [8][9][10], and hyperbolic manifolds that fiber over the circle with fiber a punctured torus [11].…”

Section: Introductionmentioning

confidence: 99%