Dehn surgery on complicated fibered knots in the 3-sphere

Thompson, Abigail

doi:10.48550/arxiv.1604.04902

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…2g + b ≥ 5 where g is the genus of F and b is the number of boundary components) and any n ∈ Z, there are infinitely many homeomorphisms ϕ : F → F such that d A (ϕ) > n (see Lemma 4.4). However, Schleimer has conjectured the following: Conjecture 1.5 (Schleimer [14]). For any closed connected oriented 3-manifold M , there is a constant t(M ) with the following property: if K ⊂ M is a fibered knot then the monodromy of K has translation distance in the arc complex of the fiber at most t(M ).…”

Section: Introductionmentioning

confidence: 99%

Translation distance bounds for fibered 3-manifolds with boundary

Stas

2019

Preprint

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Given Mϕ, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy ϕ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the minimal complexity of a surface with non-zero slope in M ϕ n tends to infinity as n → ∞. Additionally, we show that an infinite family of fibered hyperbolic knots has translation distance bounded above by two, satisfying a conjecture by Schleimer which postulates that this behavior should hold for all fibered knots.

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Section: Introductionmentioning

confidence: 99%

Translation distance bounds for fibered 3-manifolds with boundary

Stas

2019

Preprint

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“…Saul Schleimer has the following conjecture regarding the complexity of fibered knots in three-manifolds. Conjecture 1.1 (Schleimer [14], Thompson [15]). For any three-manifold M , there is a constant t(M ) with the following property: if K ⊂ M is a fibered knot, then the monodromy of K has complexity at most t(M ).…”

Section: Introductionmentioning

confidence: 99%

Heegaard genus and complexity of fibered knots

Cengiz¹

2019

Preprint

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We prove that if a fibered knot K with genus greater than one in a threemanifold M has a sufficiently complicated monodromy, then K induces a minimal genus Heegaard splitting P that is unique up to isotopy, and small genus Heegaard splittings of M are stabilizations of P . We provide a complexity bound in terms of the Heegaard genus of M . We also provide global complexity bounds for fibered knots in the three-sphere and lens spaces.

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“…Conjecture 1.1 (Schleimer (Private Communication), Thompson [16]). For any three-manifold 𝑀, there is a constant 𝑡(𝑀) with the following property: if 𝐾 ⊂ 𝑀 is a fibered knot, then the monodromy of 𝐾 has complexity at most 𝑡(𝑀).…”

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confidence: 99%

Heegaard genus and complexity of fibered knots

Cengiz

2022

Journal of Topology

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We prove that if a fibered knot K$K$ with genus greater than 1 in a three‐manifold M$M$ has a sufficiently complicated monodromy, then K$K$ induces a minimal genus Heegaard splitting P$P$ that is unique up to isotopy, and small genus Heegaard splittings of M$M$ are stabilizations of P$P$. We provide a complexity bound in terms of the Heegaard genus of M$M$. We also provide global complexity bounds for fibered knots in the three‐sphere and lens spaces.

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Dehn surgery on complicated fibered knots in the 3-sphere

Cited by 3 publications

References 12 publications

Translation distance bounds for fibered 3-manifolds with boundary

Translation distance bounds for fibered 3-manifolds with boundary

Heegaard genus and complexity of fibered knots

Heegaard genus and complexity of fibered knots

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