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15
On canonical triangulations of oncepunctured torus bundles and twobridge link complements
, 2006
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Links with no exceptional surgeries
 COMMENT. MATH. HELV
, 2006
"... We show that if a knot admits a prime, twist–reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non–trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combina ..."
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Cited by 17 (9 self)
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We show that if a knot admits a prime, twist–reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non–trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.
Cusp areas of Farey manifolds and applications to knot theory
, 2008
"... We find explicit, combinatorial estimates for the cusp areas of once–punctured torus bundles, 4–punctured sphere bundles, and 2–bridge link complements. Applications include volume estimates for the hyperbolic 3manifolds obtained by Dehn filling these bundles, for example estimates on the volume of ..."
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Cited by 9 (4 self)
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We find explicit, combinatorial estimates for the cusp areas of once–punctured torus bundles, 4–punctured sphere bundles, and 2–bridge link complements. Applications include volume estimates for the hyperbolic 3manifolds obtained by Dehn filling these bundles, for example estimates on the volume of closed 3–braid complements in terms of the complexity of the braid word. We also relate the volume of a closed 3–braid to certain coefficients of its Jones polynomial.
Commensurators of cusped hyperbolic manifolds
 Experiment. Math
"... Abstract. This paper describes a general algorithm for finding the commensurator of a nonarithmetic hyperbolic manifold with cusps, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decomp ..."
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Cited by 5 (0 self)
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Abstract. This paper describes a general algorithm for finding the commensurator of a nonarithmetic hyperbolic manifold with cusps, and for deciding when two such manifolds are commensurable. The method is based on some elementary observations regarding horosphere packings and canonical cell decompositions. For example, we use this to find the commensurators of all nonarithmetic hyperbolic oncepunctured torus bundles over the circle. For hyperbolic 3manifolds, the algorithm has been implemented using Goodman’s computer program Snap. We use this to determine the commensurability classes of all cusped hyperbolic 3manifolds triangulated using at most 7 ideal tetrahedra, and for the complements of hyperbolic knots and links with up to 12 crossings. 1.
TRIANGULATED CORES OF PUNCTUREDTORUS GROUPS
, 2006
"... We show that the interior of the convex core of a quasifuchsian puncturedtorus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an EpsteinPenner con ..."
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Cited by 5 (4 self)
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We show that the interior of the convex core of a quasifuchsian puncturedtorus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an EpsteinPenner convex hull construction in Minkowski space. The result extends to certain nonquasifuchsian puncturedtorus groups, and in fact to all of them if a strong version of the Pleating Lamination Conjecture is true.
CANONICAL TRIANGULATIONS OF DEHN FILLINGS
, 2008
"... 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, ..."
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Cited by 3 (1 self)
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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.
On hyperbolic oncepuncturedtorus bundles
 II, Geom. Dedicata
"... Abstract. To each oncepuncturedtorus bundle, Tϕ, over the circle with pseudoAnosov monodromy ϕ, there are associated two tessellations of the complex plane: one, ∆(ϕ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra ..."
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Abstract. To each oncepuncturedtorus bundle, Tϕ, over the circle with pseudoAnosov monodromy ϕ, there are associated two tessellations of the complex plane: one, ∆(ϕ), is (the projection from ∞ of) the triangulation of a horosphere at ∞ induced by the canonical decomposition into ideal tetrahedra, and the other, CW(ϕ), is a fractal tessellation given by the CannonThurston map of the fiber group switching back and forth between gray and white each time it passes through ∞. In this paper, we study the relation between ∆(ϕ) and CW(ϕ). Dedicated to Prof. Akio Kawauchi on the occasion of his 60th birthday