Bordered Floer homology and existence of incompressible tori in homology spheres

Eftekhary, Eaman

doi:10.1112/s0010437x18007054

Cited by 17 publications

(18 citation statements)

References 26 publications

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“…Furthermore, we have to require that the Uhlenbeck compactification goes through with the perturbation we have in mind. It is shown in [9, Section 5.5] that both hold for the function Γ built from holonomy perturbations as described in Equation (10). One essential feature is that the holonomy perturbation term appearing in the flow equation is uniformly bounded.…”

Section: 2mentioning

confidence: 99%

Toroidal homology spheres and SU(2)-representations

Lidman,

Pinzón-Caicedo,

Zentner

2021

Preprint

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Section: 2mentioning

confidence: 99%

Toroidal homology spheres and SU(2)-representations

Lidman,

Pinzón-Caicedo,

Zentner

2021

Preprint

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“…are true for non-trivial reasons that will not be discussed here and are not relevant for the purposes of this paper. A detailed discussion of this claim appears in [2]. Subsection 3.3 ′ : Some properties of the maps f • (K) and f • (K) Our first observation is that changing the orientation of the knot K , and correspondingly that of K 1 and K 0 , corresponds to changing the markings u, v, w with u, v, w in Figure 1.…”

Section: The Involution Of Hfkmentioning

confidence: 99%

“…However, the arguments of this paper are not sufficient for showing this and the corrections of this note are thus very crucial. The issue is further discussed in the sequel [2].…”

mentioning

confidence: 99%

Correction to the article: Floer homology and splicing knot complements

Eftekhary

2017

Preprint

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Correction to the article: Floer homology and splicing knot complements EAMAN EFTEKHARYThis note corrects the mistakes in the splicing formulas of the paper "Floer homology and splicing knot complements" [1]. The mistakes are the result of the incorrect assumption that for a knot K inside a homology sphere Y , the involution on HFK(K) which corresponds to moving the basepoints by one full twist around K is trivial. The correction implicitly involves considering the contribution from this (possibly non-trivial) involution in a number of places.

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“…If a knot in the Poincaré homology sphere is doubly primitive, then it is surgery dual to one of the Tange knots or one of the Hedden knots.1 The manifolds S 3 , P, and its mirror are the only homology 3-spheres with finite fundamental group (by Perelman) and the only known irreducible L-space homology 3-spheres, e.g. [Eft09,Eft15].2 This threshold may be lower for knots in homology spheres with τ (K) < g(K).…”

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confidence: 99%

“…1 The manifolds S 3 , P, and its mirror are the only homology 3-spheres with finite fundamental group (by Perelman) and the only known irreducible L-space homology 3-spheres, e.g. [Eft09,Eft15].…”

mentioning

confidence: 99%

The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two

Baker

2020

Pacific J. Math.

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We exhibit an infinite family of knots in the Poincaré homology sphere with tunnel number 2 that have a lens space surgery. Notably, these knots are not doubly primitive and provide counterexamples to a few conjectures. In the appendix, it is shown that hyperbolic knots in the Poincaré homology sphere with a lens space surgery has either no symmetries or just a single strong involution. − 3n 2 +n+1 −3n+2 . Through double branched coverings and the Montesinos Trick, the arc κ n lifts to a knot K n in the Poincaré homology sphere on which an integral surgery yields the lens space L(3n 2 + n + 1, −3n + 2).As presented, the arc κ n lies in a bridge sphere that presents P (−2, 3, 5) as a 3-bridge link. Thus the knot K n lies in a genus 2 Heegaard surface for the Poincaré homology sphere P. Nevertheless, as we shall see, generically K n does not have tunnel number one.Theorem 1. For each integer n, the knot K n in the Poincaré homology sphere P has an integral surgery to the lens space L(3n 2 + n + 1, −3n + 2). If |n| ≥ 4, then K n has tunnel number 2.Previously, the only known knots in P admitting a surgery to a lens space are surgery duals to Tange knots [Tan09a] and certain Hedden knots [Hed11, Ras07, Bak14b]. (See Section 5 for a discussion of the Hedden knots.) These knots in P are all doubly primitive, they may be presented as curves in a genus 2 Heegaard splitting that represent a generator in π 1 of each handlebody bounded by the Heegaard surface. We see this as Tange knots and Hedden knots all admit descriptions by doubly pointed genus 1 Heegaard diagrams.(Tange knots are all simple knots in lens spaces and Hedden knots are "almost simple".) Equivalently, they are all 1-bridge knots in their lens spaces. Any integral surgery on a 1-bridge knot in a lens space naturally produces a 3-manifold with a genus 2 Heegaard surface in which the dual knot sits as a doubly primitive knot.The Berge Conjecture posits that any knot in S 3 with a lens space surgery is a doubly primitive knot [Ber], cf. [Gor91, Question 5.5]. It is also proposed that any knot in S 1 × S 2 with a lens space surgery is a doubly primitive knot [BBL13, Conjecture 1.1], [Gre13, Conjecture 1.9]. By analogy and due to a sense of simplicity 1 , one may suspect that any knot in P with a lens space surgery should also be doubly primitive. However this is not the case. Since our knots K n have tunnel number 2 for |n| ≥ 4, they cannot be doubly primitive.Corollary 2. There are knots in the Poincaré homology sphere with lens space surgeries that are not doubly primitive. That is, the analogy to the Berge Conjecture fails for the Poincaré homology sphere.Corollary 3. Conjecture 1.10 of [Gre13] is false. For example, the lens space L(5, 1) contains a knot K * 1 with a surgery to P but does not contain a Tange knot or a Hedden knot with a P surgery.Proof. For n = 1, surgery on K 1 produces L(5, −1) as shown in Figure 1. A quick check of [Tan09a, Table 2] shows that L(5, −1) contains no Tange knots. The Hedden knots with surgery to ±P are homologous to Berge ...

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Bordered Floer homology and existence of incompressible tori in homology spheres

Cited by 17 publications

References 26 publications

Toroidal homology spheres and SU(2)-representations

Toroidal homology spheres and SU(2)-representations

Correction to the article: Floer homology and splicing knot complements

The Poincaré homology sphere, lens space surgeries, and some knots with tunnel number two

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