All integral slopes can be Seifert fibered slopes for hyperbolic knots

Motegi, Kimihiko; Song, Hyun-Jong

doi:10.2140/agt.2005.5.369

Cited by 4 publications

(4 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using relations ( 12), (15), and ( 16) one sees that this can only happen for (p, q) = (0, 1) and (p, q) = (−1, −2), but det 1 1 p q = 1, so (p, q) = (0, 1), as required. Even if not necessary, note that we have shown the equality N (−3, 1, 1) = (K, 1), which is coherent with (29) and (23).…”

Section: Naturality Of Basessupporting

confidence: 82%

See 1 more Smart Citation

Dehn Filling of the "Magic" 3-manifold

Martelli¹,

Petronio²

2006

Communications in Analysis and Geometry

179

View full text Add to dashboard Cite

We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume.Among other consequences of this classification, we mention the following:• for every integer n, we can prove that there are infinitely many hyperbolic knots in S 3 having exceptional surgeries {n, n + 1, n + 2, n + 3}, with n + 1, n + 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds.• we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements.• we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements.• we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling. IntroductionWe study in this paper the Dehn fillings of the complement N of the chain link with three components in S 3 , shown in figure 1. The hyperbolic structure of N was first constructed by Thurston in his notes [28], and it was also noted there that the volume of N is particularly small. The relevance of N to three-dimensional topology comes from the fact that by filling N , one gets most of the hyperbolic manifolds known and most of the interesting non-hyperbolic fillings of cusped hyperbolic manifolds. For these reasons N was called the "magic manifold" by Gordon and Wu [14,17]. It appears as M 6 3 1 in [6] and it is the hyperbolic manifold with three cusps of smallest known volume and of smallest complexity [1]. (We refer here to the complexity defined by Matveev in [23], and we mean that N has an ideal triangulation with six tetrahedra, while all other hyperbolic manifolds 969

show abstract

Section: Naturality Of Basessupporting

confidence: 82%

“…Motegi and Song recently proved that for every n there is a hyperbolic knot whose n-surgery is small Seifert [23], and Teragaito proved the same result for toroidal surgeries [26]. Theorem 1 exhibits infinitely many examples for each n. Other knots in S 3 with Seifert surgeries are also described in [7,8,21].…”

mentioning

confidence: 90%

Dehn Filling of the "Magic" 3-manifold

Martelli¹,

Petronio²

2006

Communications in Analysis and Geometry

179

View full text Add to dashboard Cite

show abstract

“…(Proposition 3.9) Proposition 3.9 and its proof imply the following theorem, which generalizes a previous result in [18]. Theorem 3.13.…”

Section: Seiferters For Seifert Surgeries On a Trefoil Knotsupporting

confidence: 70%

“…(Proposition 3.9) Proposition 3.9 and its proof imply the following theorem, which generalizes a previous result in [18]. (1) For any integer m, there is a hyperbolic knot K such that (K, m), (K, m+ 1), (K, m + 2) are small Seifert fibered surgeries.…”

Section: Seiferters For Seifert Surgeries On a Trefoil Knotsupporting

confidence: 56%

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

Deruelle

Miyazaki

Motegi

2014

Bol. Soc. Mat. Mex.

Self Cite

View full text Add to dashboard Cite

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

show abstract

Exceptional Dehn Surgeries on Some Infinite Series of Hyperbolic Knots and Links

Cavicchioli,

Spaggiari

2024

Mediterr. J. Math.

View full text Add to dashboard Cite

We study closed connected orientable 3-manifolds obtained by Dehn surgery along the oriented components of a link, introduced and considered by Motegi and Song (2005) and Ichihara et al. (2008). For such manifolds, we find a finite balanced group presentation of the fundamental group and describe exceptional surgeries. This allows us to construct an infinite family of tunnel number one strongly invertible hyperbolic knots with three parameters, which admit toroidal surgeries and Seifert fibered surgeries. Among the obtained results, we mention that for every integer $$n >5$$ n > 5 there are infinitely many hyperbolic knots in the 3–sphere, whose $$(n-2)$$ ( n - 2 ) and $$(n+1)$$ ( n + 1 ) -surgeries are toroidal, and $$(n-1)$$ ( n - 1 ) and n-surgeries are Seifert fibered.

show abstract

All integral slopes can be Seifert fibered slopes for hyperbolic knots

Cited by 4 publications

References 29 publications

Dehn Filling of the "Magic" 3-manifold

Dehn Filling of the "Magic" 3-manifold

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

Exceptional Dehn Surgeries on Some Infinite Series of Hyperbolic Knots and Links

Contact Info

Product

Resources

About