Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions

Eudave-Muñoz, Mario; Jasso, Edgar; Miyazaki, Katura; Motegi, Kimihiko

doi:10.1016/j.topol.2015.05.016

Cited by 8 publications

(10 citation statements)

References 21 publications

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“…Note that (−1)-surgery on T w(n 0 ) yields a Seifert fiber space over S 2 (2, 3, |6n 0 − 1|); Figure 5.9 gives a pictorial proof of this fact. On the other hand, K p0 (−1) is a T -3, 2 c In [7], using the Montesinos trick, we find an infinite family of small Seifert fibered surgeries on strongly invertible hyperbolic knots which do not arise from the primitive/Seifert-fibered construction. The same family is also obtained from (T −3 particular, for pointing out an error in the proof of Proposition 4.4, and would like to thank Mario Eudave-Muñoz for useful comments.…”

Section: Strongly Invertible Knots That Do Not Arise From the Primitimentioning

confidence: 95%

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

Deruelle

Miyazaki

Motegi

2014

Bol. Soc. Mat. Mex.

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A Seifert surgery is a pair (K , m) of a knot K in S 3 and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K , m) yields Seifert surgeries. We study Seifert surgeries obtained from those on a trefoil knot by twisting along their seiferters. Although Seifert surgeries on a trefoil knot are the most basic ones, this family is rich in variety. For any m = −2 it contains a successive triple of Seifert surgeries (K , m), (K , m + 1), (K , m + 2) on a hyperbolic knot K , e.g. 17-, 18-, 19-surgeries on the (−2, 3, 7) pretzel knot. It contains infinitely many Seifert surgeries on strongly invertible hyperbolic knots none of which arises from the primitive/Seifertfibered construction, e.g. (−1)-surgery on the (3, −3, −3) pretzel knot.Keywords Dehn surgery · Hyperbolic knot · Seifert fiber space · Trefoil knot · seiferter · Seifert surgery network Dedicated to Fico González-Acuña on his 70th birthday.A. Deruelle

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Section: Strongly Invertible Knots That Do Not Arise From the Primitimentioning

confidence: 95%

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

Deruelle

Miyazaki

Motegi

2014

Bol. Soc. Mat. Mex.

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“…Proof of Theorem 8.1. We begin by recalling the following result which is a combination of Propositions 3.2, 3.7 and 3.11 in [16].…”

Section: Hyperbolic L-space Knots With Tunnel Number Greater Than Onementioning

confidence: 96%

“…(c a , (−1)−twist) → (c b , 1−twist) → (c a , n−twist) Then K n,0 = K(2, −n, 1, 0) in [16,Proposition 4.11]. See Figure 8 (1) {K n,0 } n∈Z is a set of mutually distinct hyperbolic L-space knots with tunnel number two.…”

Section: Hyperbolic L-space Knots With Tunnel Number Greater Than Onementioning

confidence: 99%

L-space surgery and twisting operation

Motegi

2016

Algebr. Geom. Topol.

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A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, i.e. a rational homology 3-sphere with the smallest possible Heegaard Floer homology. Given a knot K, take an unknotted circle c and twist K n times along c to obtain a twist family { K_n }. We give a sufficient condition for { K_n } to contain infinitely many L-space knots. As an application we show that for each torus knot and each hyperbolic Berge knot K, we can take c so that the twist family { K_n } contains infinitely many hyperbolic L-space knots. We also demonstrate that there is a twist family of hyperbolic L-space knots each member of which has tunnel number greater than one.Comment: The final version, accepted for publication by Algebr. Geom. Topo

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“…However we will see that surgery along the framing induced by the Heegaard surface produces a toroidal manifold which, according to Kobayashi's classification [Kob84], does not have Heegaard genus 2. (The same scheme was recently employed in [EMJMM14] to demonstrate a family of strongly invertible knots in S 3 with a Seifert fibered surgery and tunnel number 2. )…”

Section: Proofsmentioning

confidence: 97%

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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Seifert fibered surgeries on strongly invertible knots without primitive/Seifert positions

Cited by 8 publications

References 21 publications

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

Neighbors of Seifert surgeries on a trefoil knot in the Seifert Surgery Network

L-space surgery and twisting operation

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

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