Networking Seifert surgeries on knots II: The Berge's lens surgeries

Deruelle, Arnaud; Miyazaki, Katura; Motegi, Kimihiko

doi:10.1016/j.topol.2008.10.012

Cited by 10 publications

(29 citation statements)

References 13 publications

(31 reference statements)

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“…Since c is a seiferter for (T 3p+1,2p+1 , (3p+ 1)(2p+ 1)) and (4p+ 1) 2 > 2(3p+ 1)(2p+ 1), Theorem 1.7 shows that K(3p + 1, 2p + 1; 4p + 1, n) is an L-space knot for all integers n. Let us observe that K(3p + 1, 2p + 1; 4p + 1, n) is a hyperbolic knot if |n| > 3. Figure 44 in [11] shows that n-twist converts (T 3p+1,2p+1 , (3p + 1)(2p + 1)) into a Seifert surgery which is not a lens space surgery if |n| ≥ 2. Hence c becomes a degenerate fiber in T 3p+1,2p+1 ((3p+1)(2p+1)) [13, Lemma 5.6(1)], and Corollary 3.21(3) in [13] shows that the link T 3p+1,2p+1 ∪ c is hyperbolic.…”

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

“…Now the result follows from [13, Proposition 5.11 (2) • K(3p + 2, 2p + 1; 4p + 3, n) (p > 0). As above, we follow the argument in [11,Subsection 8.3], but we need to take the mirror image at the end. Take a torus knot k = T −p−1,2p+1 on a genus two Heegaard surface of S 3 , unknotted circles α ′ and c ′ as shown in Figure 5.5.…”

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

“…As above we denote the image of c ′ after (−1)twist along α ′ by the same symbol c ′ ; the linking number between c ′ and T −3p−2,2p+1 is 4p + 3. Note that (−1)-twist along c ′ converts T −3p−2,2p+1 into a Berge knot Sporc[p] as shown in [11,Subsection 8.3]. Then Lemma 8.6 in [11] shows that c ′ is a seiferter for a lens space surgery (Sporc[p], −22p 2 − 31p − 11) = (Sporc[p], (−3p − 2)(2p + 1) − (4p + 3) 2 ).…”

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

“…Note that (−1)-twist along c ′ converts T −3p−2,2p+1 into a Berge knot Sporc[p] as shown in [11,Subsection 8.3]. Then Lemma 8.6 in [11] shows that c ′ is a seiferter for a lens space surgery (Sporc[p], −22p 2 − 31p − 11) = (Sporc[p], (−3p − 2)(2p + 1) − (4p + 3) 2 ). Thus c ′ is also a seiferter for (T −3p−2,2p+1 , (−3p− 2)(2p+ 1)).…”

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

“…In the above theorem, even when ℓ 2 < 2pq, K n (n < −1) may be an L-space knot; see [41]. In Sections 5, 6 and 7 we will exploit seiferter technology developed in [13,11,12] to give a partial answer to Question 1.1. Even though Theorem 1.7 treats a special kind of Seifert surgeries, it offers many applications.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

Section: L-space Twisted Torus Knotsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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L-space surgery and twisting operation

Motegi

2016

Algebr. Geom. Topol.

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Lens Space Surgeries Along Certain 2-Component Links Related With Park’s rational Blow Down, and Reidemeister-Turaev Torsion

Kadokami

Yamada²

2013

J. Aust. Math. Soc.

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We study lens space surgeries along two different families of 2-component links, denoted by A m,n and B p,q , related with the rational homology 4-ball used in J. Park's (generalized) rational blow down. We determine which coefficient r of the knotted component of the link yields a lens space by Dehn surgery. The link A m,n yields a lens space only by the known surgery with r = mn and unexpectedly with r = 7 for (m, n) = (2, 3). On the other hand, B p,q yields a lens space by infinitely many r. Our main tool for the proof are the Reidemeister-Turaev torsions, that is, Reidemeister torsions with combinatorial Euler structures. Our results can be extended to the links whose Alexander polynomials are same as those of A m,n and B p,q .surgery if the result is a lens space. We study lens space surgeries along the links A m,n and B p,q , fixing the surgery coefficient of K 2 as 0 except in Section 6.4 and Section 7. Our convention on lens spaces is 'L(a, b) is the result of −a/b-surgery along the trivial knot'.The result of (r, 0)-surgery along a 2-component link L = K 1 ∪ K 2 in S 3 (as ∂B 4 ) such that K 2 is an unknot and r ∈ Z bounds a 4-manifold by attaching a 1-handle along K 2 , and a 2-handle along K 1 with a framing r, to a 0-handle B 4 along S 3 . We denote the 4-manifold by W 4 (L; r,0) following Akbulut [Ak] (see also [GS, Kir]). Then π 1 (W 4 (L; r,0)) Z/|l|Z and H 1 (∂W 4 (L; r,0); Z) Z/l 2 Z, where l is the linking number of K 1 and K 2 , and W 4 (L; r,0) is a rational homology 4-ball if and only if l 0. We also note that K 1 can be regarded as a knot in S 1 × S 2 .We explain a background of our targets A m,n and B p,q . Park [Pa] discussed generalized rational blow down, which is an operation on a 4-manifold cutting a certain submanifold C p,q and pasting a 4-manifold W p,q along ∂C p,q ∂W p,q . The 4manifold W p,q is a rational homology 4-ball that is characterized by π 1 (W p,q ) Z/pZ and ∂W p,q L(p 2 , pq − 1) (see [CH, FS]). To the best of the authors' knowledge, it is unknown whether the diffeomorphism type of W p,q is determined by these properties. A lens space surgery along B p,q whose result is L(p 2 , pq − 1) is often [2] Lens space surgeries along certain 2-component links 79 T. Kadokami and Y. Yamada [3] [4]Lens space surgeries along certain 2-component links 81 T. Kadokami and Y. Yamada [9]

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Networking Seifert Surgeries on Knots IV: Seiferters and Branched Coverings

Deruelle¹,

Eudave-Muñoz²,

Miyazaki³

et al. 2013

Geometry and Topology Down Under

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A Seifert surgery is an integral surgery on a knot in S 3 producing a Seifert fiber space M which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pairs of seiferters". One problem of the network is whether there is a path from each vertex to a vertex on a torus knot, the most basic Seifert surgery. We give a method to find seiferters and annular pairs of seiferters for Seifert surgeries obtained by taking two-fold branched covers of tangles. Concerning three infinite families of Seifert surgeries obtained by the second author via branched covers, we find explicit paths in the network from such surgeries to Seifert surgeries on torus knots. 1991 Mathematics Subject Classification. Primary 57M25, 57M50 Secondary 57N10.

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Networking Seifert surgeries on knots II: The Berge's lens surgeries

Cited by 10 publications

References 13 publications

L-space surgery and twisting operation

L-space surgery and twisting operation

Lens Space Surgeries Along Certain 2-Component Links Related With Park’s rational Blow Down, and Reidemeister-Turaev Torsion

Networking Seifert Surgeries on Knots IV: Seiferters and Branched Coverings

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