Satellite knots and L-space surgeries

Hom, Jennifer

doi:10.1112/blms/bdw044

Cited by 7 publications

(11 citation statements)

References 15 publications

(25 reference statements)

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“…This confirms a conjecture of Hom, Lidman, and Vafaee [20,Conjecture 1.7]. It has subsequently been used by Baker and Motegi [4] to show that the pattern knot must be a braid in the solid torus (see also Hom [19]).…”

Section: P( Hf (M S))supporting

confidence: 78%

Bordered Floer homology for manifolds with torus boundary via immersed curves

Hanselman¹,

Rasmussen²,

Watson³

2016

Preprint

154

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This paper gives a geometric interpretation of bordered Heegaard Floer homology for manifolds with torus boundary. If M is such a manifold, we show that the type D structure CFD(M ) may be viewed as a set of immersed curves decorated with local systems in ∂M . These curves-with-decoration are invariants of the underlying three-manifold up to regular homotopy of the curves and isomorphism of the local systems. Given two such manifolds and a homeomorphism h between the boundary tori, the Heegaard Floer homology of the closed manifold obtained by gluing with h is obtained from the Lagrangian intersection Floer homology of the curve-sets. This machinery has several applications: We establish that the dimension of HF decreases under a certain class of degree one maps (pinches) and we establish that the existence of an essential separating torus gives rise to a lower bound on the dimension of HF . In particular, it follows that a prime rational homology sphere Y with HF (Y ) < 5 must be geometric. Other results include a new proof of Eftekhary's theorem that L-space homology spheres are atoroidal; a complete characterisation of toroidal L-spaces in terms of gluing data; and a proof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

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Section: P( Hf (M S))supporting

confidence: 78%

Bordered Floer homology for manifolds with torus boundary via immersed curves

Hanselman¹,

Rasmussen²,

Watson³

2016

Preprint

154

View full text Add to dashboard Cite

show abstract

“…Thus, under such an assumption, we affirmatively answer the first part of [4, Question 22]. More broadly, without the need to assume the conjecture, together Proposition 6.5 and Corollary 6.6 affirmatively answer the first part of [27,Question 1.8].…”

Section: Lemma 62 Assume That K N Is An L-space Knot For Infinitely M...mentioning

confidence: 52%

“…Using Theorem (more precisely, assuming its veracity), Hom [, Proposition 3.3] proves the following result. Although she states the result in the case where

P (K)

is a positive L‐space knot, the case when

P (K)

is a negative L‐space knot follows immediately since

\bar{P (K)} = \bar{P} false(\bar{K} false)

.…”

Section: Satellite L‐space Knotsmentioning

confidence: 99%

Seifert vs. slice genera of knots in twist families and a characterization of braid axes

Baker

Motegi

2019

Proceedings of London Math Soc

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Twisting a knot K in S 3 along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. We prove that if the ratio of the Seifert genus to the slice genus for knots in a twist family limits to 1, then the winding number of K about c equals either zero or the wrapping number. As a key application, if {Kn} or the mirror twist family {Kn} contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that c is a braid axis of K if and only if both {Kn} and {Kn} each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for {Kn} to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore. Contents

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“…Satellite links whose patterns are closed braids and whose companions are fibred are a particularly interesting class to investigate as, for instance, all satellite L-space knots in S 3 fall into this category ( [BM,Theorem 7.3,Theorem 7.4]; also see [HRW,Theorem 35] and [Hom,Proposition 3.3]). Theorem 1.14 and Corollary 1.15 verify special cases of Conjecture 1.13.…”

Section: Introductionmentioning

confidence: 99%

“…In the case that both pattern and companion are hyperbolic, and the fractional Dehn twist coefficient of the pattern braid is nonnegative (cf. [Hom,Question 1.8]), Theorem 1.16 shows that Σ n (K) is excellent whenever n is relatively prime to the braid index of the pattern.…”

Section: Introductionmentioning

confidence: 99%

Taut foliations in branched cyclic covers and left-orderable groups

Boyer

2019

Trans. Amer. Math. Soc.

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We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology 3-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible 3-manifolds containing a genus 1 fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology 3-sphere admits a co-oriented taut foliation and has left-orderable fundamental group, even if the surgered manifold does not, and that the same holds for many branched covers of satellite knots with braided patterns. A key fact used in our proofs is that the Euler class of a universal circle representation associated to a co-oriented taut foliation coincides with the Euler class of the foliation's tangent bundle. Though known to experts, no proof of this important result has appeared in the literature. We provide such a proof in the paper.Conjecture 1.1 (Conjecture 1 in [BGW], Conjecture 5 in [Ju]). Assume that M is a closed, connected, irreducible, orientable 3-manifold. Then the following statements are equivalent.(1) M is not a Heegaard Floer L-space,(2) M admits a co-orientable taut foliation,(3) π 1 (M ) is left-orderable.

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Satellite knots and L-space surgeries

Abstract: We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns that produce satellite L-space knots.

Cited by 7 publications

References 15 publications

Bordered Floer homology for manifolds with torus boundary via immersed curves

Bordered Floer homology for manifolds with torus boundary via immersed curves

Seifert vs. slice genera of knots in twist families and a characterization of braid axes

Taut foliations in branched cyclic covers and left-orderable groups

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