Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components

Lee, Sangyop

doi:10.1016/j.top.2007.03.003

Cited by 9 publications

(8 citation statements)

References 32 publications

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“…By Proposition 3.1, the exterior M Ls = M L is hyperbolic, and thus we can apply Theorem 1.1 of [17], which asserts that K s is hyperbolic whenever |s| > 5. Figure 1: An example of a link L s .…”

Section: Genus Zero Bridge Numbers Of Some Twisted Torus Knotsmentioning

confidence: 99%

Bridge Spectra of Twisted Torus Knots

Bowman

Taylor

Zupan

2014

Int Math Res Notices

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We compute the genus zero bridge numbers and give lower bounds on the genus one bridge numbers for a large class of sufficiently generic hyperbolic twisted torus knots. As a result, the bridge spectra of these knots have two gaps which can be chosen to be arbitrarily large, providing the first known examples of hyperbolic knots exhibiting this property. In addition, we show that there are Berge and Dean knots with arbitrarily large genus one bridge numbers, and as a result, we give solutions to problems of Eudave-Muñoz concerning tunnel number one knots.

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Section: Genus Zero Bridge Numbers Of Some Twisted Torus Knotsmentioning

confidence: 99%

Bridge Spectra of Twisted Torus Knots

Bowman

Taylor

Zupan

2014

Int Math Res Notices

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“…By [6], ∂M is a single torus or two tori. However, the latter case does not happen by [14]. For the former, Theorem 1.1 gives a contradiction.…”

Section: Proofs Of Main Resultsmentioning

confidence: 86%

“…Then, we showed that ∂M is a single torus or two tori [6]. By [14], the latter case happens only when M is the Whitehead sister link exterior. In this case, it is well known that each surgered manifold contains an essential torus meeting the core of the attached solid torus just twice [13].…”

Section: Proofs Of Main Resultsmentioning

confidence: 93%

Toroidal Dehn fillings on large hyperbolic 3-manifolds

Teragaito¹

2006

Communications in Analysis and Geometry

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We show that if a hyperbolic 3-manifold M with a single torus boundary admits two Dehn fillings at distance 5, each of which contains an essential torus, then M is a rational homology solid torus, which is not large in the sense of Wu. Moreover, one of the surgered manifolds contains an essential torus which meets the core of the attached solid torus minimally in at most two points. This completes the determination of best possible upper bounds for the distance between two exceptional Dehn fillings yielding essential small surfaces in all ten cases for large hyperbolic 3-manifolds.2000 Mathematics Subject Classification. Primary 57M50.

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“…Neither X (π) nor X (τ ) is hyperbolic (the former contains a Möbius bandB and the latter contains an essential torus T ) and (π, τ ) = 5, so it follows from [Lee 2007, Theorem 1.1] that X is the exterior of the Whitehead sister link. Hence M is the result of a Dehn filling on the link exterior.…”

Section: The Case P =mentioning

confidence: 99%

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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Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components

Cited by 9 publications

References 32 publications

Bridge Spectra of Twisted Torus Knots

Bridge Spectra of Twisted Torus Knots

Toroidal Dehn fillings on large hyperbolic 3-manifolds

Klein bottle and toroidal Dehn fillings at distance 5

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