Counting genus one fibered knots in lens spaces

Baker, Kenneth L.

doi:10.1307/mmj/1409932633

Cited by 12 publications

(14 citation statements)

References 18 publications

(21 reference statements)

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“…By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto [12] and Baker [2], [3].…”

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confidence: 73%

“…H 1 (T ),α = −2α − 3β andβ = 3α + 4β. The monodromy of the third fiber is represented in GL 2 (Z) as −− 4 in GL 2 (Z).This finishes a reproof of the Baker's results in[3]. monodromy of the third fiber in L(4, 3)…”

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confidence: 66%

“…In this paper, our goal is to give another proof of the following theorems already obtained in [2], [3], [12].…”

Section: Fibered Annulusmentioning

confidence: 93%

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Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

Sekino

2018

Topology and its Applications

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We give a necessary and sufficient condition for a simple closed curve on the boundary of a genus two handlebody to decompose the handlebody into T × I (T is a torus with one boundary component). We use this condition to decide whether a simple closed curve on a genus two Heegaard surface is a GOF-knot (genus one fibered knot) which induces the Heegaard splitting. By using this, we determine the number and the positions with respect to the Heegaard splittings of GOF-knots in the 3-manifolds with reducible genus two Heegaard splittings. This is another proof of results of Morimoto [12] and Baker [2], [3].

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confidence: 73%

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confidence: 66%

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Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

Sekino

2018

Topology and its Applications

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“…Proof. This can be seen in the spirit of [Bak14a] which relates genus one fibered knot to axes of closed 3-braids as follows: Noting that since P has a unique genus 2 Heegaard splitting [BO91], P (−2, 3, 5) is the only 3-bridge link whose double branched cover gives P. (In fact P is the double branched cover of no other link.) Since P (−2, 3, 5) is isotopic to the (3, 5)-torus knot, its 3-braid axis A lifts to a genus one fibered knot J ⊂ P. By [BM93, The Classification Theorem] and [Bak14a, Lemma 3.8], for instance, A is the only 3-braid axis for P (−2, 3, 5) up to isotopy of unoriented links.…”

Section: Proof Of Theoremmentioning

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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“…We will explain the background of the fact and formulate a proof of the fact in Section 3. Some equivalent classes of genus one fibered knots in lens spaces are studied by Morimoto [27], and all such classes are completely classified by Baker [2]. Following the classification, by applying Proposition 3.3, we determine the genus one fibered knots in lens spaces on whose all integral Dehn surgeries yield closed 3-manifolds with left-orderable fundamental groups.…”

Section: Introductionmentioning

confidence: 99%

Integral left-orderable surgeries on genus one fibered knots

Ichihara,

Nakae

2020

Preprint

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Counting genus one fibered knots in lens spaces

Cited by 12 publications

References 18 publications

Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

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

Integral left-orderable surgeries on genus one fibered knots

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