Toroidal Dehn fillings on large hyperbolic 3-manifolds

Teragaito, Masakazu

doi:10.4310/cag.2006.v14.n3.a6

Cited by 4 publications

(1 citation statement)

References 17 publications

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“…Others have made similar observations. Notably, Teragaito [29] conjectured that integral exceptional surgeries occur as a sequence of consecutive integers. Dunfield [10] showed that, for small knots, if surgery along slope r yields a manifold with cyclic fundamental group, then there is a non-integral boundary slope in the interval (r − 1, r + 1).…”

Section: Introductionmentioning

confidence: 99%

Boundary slopes (nearly) bound cyclic slopes

Mattman

2005

Algebr. Geom. Topol.

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Section: Introductionmentioning

confidence: 99%

Boundary slopes (nearly) bound cyclic slopes

Mattman

2005

Algebr. Geom. Topol.

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Lens spaces and toroidal Dehn fillings

Lee

2009

Math. Z.

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Klein bottle and toroidal Dehn fillings at distance 5

Lee¹

2010

Pacific J. Math.

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We determine all hyperbolic 3-manifolds M such that M(π ) contains a Klein bottle, M(τ ) contains an essential torus, and (π, τ ) = 5. As a corollary, we prove that if a hyperbolic 3-manifold M has two slopes π and τ on its boundary torus such that M(π ) is a lens space containing a Klein bottle and M(τ ) is toroidal, then (π, τ ) ≤ 4. IntroductionLet M be a compact, connected, orientable 3-manifold with a torus boundary component ∂ 0 M. A slope on ∂ 0 M is the isotopy class of an essential simple closedis obtained from M by gluing a solid torus V γ along ∂ 0 M so that γ bounds a meridional disk of V γ . For two slopes γ 1 , γ 2 on ∂ 0 M, denote by (γ 1 , γ 2 ) the distance between the slopes, which is their geometric intersection number.We shall say that a 3-manifold M is hyperbolic if M with its boundary tori removed admits a complete hyperbolic structure with totally geodesic boundary. A Dehn filling on M is said to be exceptional if it produces a nonhyperbolic 3-manifold, which is either reducible, boundary-reducible, annular, toroidal, or a small Seifert fiber space. It is a well-known theorem of Thurston that there are only finitely many exceptional Dehn fillings on each boundary torus of M.Gordon and Wu [2008] determined all hyperbolic 3-manifolds admitting two toroidal Dehn fillings at distance 4 or 5. In this paper, we determine all hyperbolic 3-manifolds M admitting two Dehn fillings at distance 5, one of which yields a Klein bottle, the other yielding an essential torus.Following [Martelli and Petronio 2006], we use N to denote the magic manifold, the exterior of the chain link with three components in S 3 , shown in Figure 1. Using the standard meridian-longitude framing on each boundary component of N , we identify a slope γ with a number in ‫ޑ‬ ∪ {1/0}. We denote by N (r ) the result of MSC2000: 57M50.

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Toroidal Dehn fillings on large hyperbolic 3-manifolds

Cited by 4 publications

References 17 publications

Boundary slopes (nearly) bound cyclic slopes

Boundary slopes (nearly) bound cyclic slopes

Lens spaces and toroidal Dehn fillings

Klein bottle and toroidal Dehn fillings at distance 5

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