Some applications of Gabaiʼs internal hierarchy

Ni, Yi

doi:10.1016/j.aim.2013.10.001

Cited by 4 publications

(1 citation statement)

References 32 publications

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“…We also define H i : CF + (Σ, α, η i , w) → CF + (Σ, α, η i+2 , w) by H i (x) = μ 3 (x, Θ i,i+1 , Θ i+1,i+2 ). It is standard to check (17)…”

Section: Knots In Torus Bundlesmentioning

confidence: 99%

Property G and the 4-genus

2024

Trans. Amer. Math. Soc. Ser. B

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We say a null-homologous knot K K in a 3 3 -manifold Y Y has Property G, if the Thurston norm and fiberedness of the complement of K K is preserved under the zero surgery on K K . In this paper, we will show that, if the smooth 4 4 -genus of K × { 0 } K\times \{0\} (in a certain homology class) in ( Y × [ 0 , 1 ] ) # N C P 2 ¯ (Y\times [0,1])\#N\overline {\mathbb CP^2} , where Y Y is a rational homology sphere, is smaller than the Seifert genus of K K , then K K has Property G. When the smooth 4 4 -genus is 0 0 , Y Y can be taken to be any closed, oriented 3 3 -manifold.

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“…We also define H i : CF + (Σ, α, η i , w) → CF + (Σ, α, η i+2 , w) by H i (x) = μ 3 (x, Θ i,i+1 , Θ i+1,i+2 ). It is standard to check (17)…”

Section: Knots In Torus Bundlesmentioning

confidence: 99%

Property G and the 4-genus

2024

Trans. Amer. Math. Soc. Ser. B

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Null surgery on knots in L-spaces

Vafaee

2019

Trans. Amer. Math. Soc.

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Let K be a knot in an L-space Y with a Dehn surgery to a surface bundle over S 1 . We prove that K is rationally fibered, that is, the knot complement admits a fibration over S 1 . As part of the proof, we show that if K ⊂ Y has a Dehn surgery to S 1 × S 2 , then K is rationally fibered. In the case that K admits some S 1 × S 2 surgery, K is Floer simple, that is, the rank of HF K(Y, K) is equal to the order of H1(Y ). By combining the latter two facts, we deduce that the induced contact structure on the ambient manifold Y is tight.In a different direction, we show that if K is a knot in an L-space Y , then any Thurston norm minimizing rational Seifert surface for K extends to a Thurston norm minimizing surface in the manifold obtained by the null surgery on K (i.e., the unique surgery on K with b1 > 0).It is a well-known fact that, up to isotopy, there exists a unique simple closed curve α on ∂ν(K) with the property that the surgery on K with slope α produces a manifold with the first Betti number one higher than that of Y . For simplicity of referring to this slope in the paper, we make the following definition: Definition 1.4. For a knot K ⊂ Y , let α be the unique slope on ∂ν(K) that is rationally nullhomologous in Y \ ν • (K). We call α the null slope of K in Y . We also define the null surgery on K to be Dehn filling the exterior of K in Y along the curve α and denote it Y α . Theorem 1.1 can be stated in terms of the dual knot K of L inside the L-space. More precisely, for an L-space Y , if the null surgery on K ⊂ Y results in S 1 × S 2 , then K is fibered. We generalize Theorem 1.1 by replacing S 1 × S 2 with an oriented closed three-manifold that is a surface bundle over S 1 . Compare the following theorem with [Ni07, Corollary 1.4].Theorem 1.5. Let K be a knot in an L-space Y . If the null surgery on K is a surface bundle over S 1 , then the complement of K in Y admits a fibration over S 1 .

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Four-dimensional Haken cobordism theory

Foozwell¹,

Rubinstein²

2016

Illinois J. Math.

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Cobordism of Haken n-manifolds is defined by a Haken (n+1)-manifold W whose boundary has two components, each of which is a closed Haken n-manifold. In addition, the inclusion map of the fundamental group of each boundary component to π1(W ) is injective. In this paper we prove that there are 4-dimensional Haken cobordisms whose boundary consists of any two closed Haken 3-manifolds. In particular, each closed Haken 3-manifold is the π1-injective boundary of some Haken 4-manifold.

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Some applications of Gabaiʼs internal hierarchy

Cited by 4 publications

References 32 publications

Property G and the 4-genus

Property G and the 4-genus

Null surgery on knots in L-spaces

Four-dimensional Haken cobordism theory

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