Lens spaces and toroidal Dehn fillings

Lee, Sangyop

doi:10.1007/s00209-009-0646-0

Cited by 3 publications

(1 citation statement)

References 17 publications

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“…Notice that in the case of (S, S 3 ), this is equivalent to the cabling conjecture, and in the case of (T, L), the bound is known to be either 3 or 4 [Lee11]. For a more thorough discussion of these bounds, the manifolds achieving them, and precise references, see [Gor09] and [Gor99].…”

Section: Some Exceptional Dehn Surgery Resultsmentioning

confidence: 99%

Small Seifert fibered surgery on hyperbolic pretzel knots

Meier

2014

Algebr. Geom. Topol.

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Section: Some Exceptional Dehn Surgery Resultsmentioning

confidence: 99%

Small Seifert fibered surgery on hyperbolic pretzel knots

Meier

2014

Algebr. Geom. Topol.

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Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

Boyer,

Gordon,

Zhang

2024

Memoirs of the AMS

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We show that if a hyperbolic knot manifold M M contains an essential twice-punctured torus F F with boundary slope β \beta and admits a filling with slope α \alpha producing a Seifert fibred space, then the distance between the slopes α \alpha and β \beta is less than or equal to 5 5 unless M M is the exterior of the figure eight knot. The result is sharp; the bound of 5 5 can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the α \alpha -filling contains no non-abelian free group. The proofs are divided into the four cases F F is a semi-fibre, F F is a fibre, F F is non-separating but not a fibre, and F F is separating but not a semi-fibre, and we obtain refined bounds in each case.

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Dehn fillings of knot manifolds containing essential once-punctured tori

Boyer¹,

Gordon

Zhang

2013

Trans. Amer. Math. Soc.

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Abstract. In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let M be such a knot manifold and let β be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling M with slope α produces a Seifert fibred manifold, then ∆(α, β) ≤ 5. Furthermore we classify the triples (M ; α, β) when ∆(α, β) ≥ 4. More precisely, when ∆(α, β) = 5, then M is the (unique) manifold W h(−3/2) obtained by Dehn filling one boundary component of the Whitehead link exterior with slope −3/2, and (α, β) is the pair of slopes (−5, 0). Further, ∆(α, β) = 4 if and only if (M ; α, β) is the triple (W h( −2n ± 1 n ); −4, 0) for some integer n with |n| > 1. Combining this with known results, we classify all hyperbolic knot manifolds M and pairs of slopes (β, γ) on ∂M where β is the boundary slope of an essential once-punctured torus in M and γ is an exceptional filling slope of distance 4 or more from β. Refined results in the special case of hyperbolic genus one knot exteriors in S 3 are also given.

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Lens spaces and toroidal Dehn fillings

Abstract: We show that if M is a hyperbolic 3-manifold with ∂ M a torus such that M(r 1 ) is a lens space and M(r 2 ) is toroidal, then (r 1 , r 2 ) ≤ 4.

Cited by 3 publications

References 17 publications

Small Seifert fibered surgery on hyperbolic pretzel knots

Small Seifert fibered surgery on hyperbolic pretzel knots

Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

Dehn fillings of knot manifolds containing essential once-punctured tori

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