The ?-orbifold group of a link

Boileau, Michel; Zimmermann, Bruno

doi:10.1007/bf01230281

Cited by 77 publications

(88 citation statements)

References 46 publications

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“…This contradiction finishes the proof of Lemma 9.2(1). Lemma 9.2(2) and (3) now follow from Lemma 9.1, (1) and (2). Finally, (4) follows from Lemma 8.1 applied to R.1=.…”

Section: Some Algebraic Tangle Calculationsmentioning

confidence: 85%

“…To prove (2), observe that rotating Figure 2.3 through 180 ı about an axis perpendicular to the plane of the paper shows that S.˛;ˇI ; ı/ D S.ı; Iˇ;˛/. (A rational tangle is unchanged by rotation through 180 ı about any of the three co-ordinate axes.)…”

Section: Preliminariesmentioning

confidence: 99%

“…We are now in the context of Theorem 5.2. Possibilities (1), (2), and (3) in the conclusion of Theorem 5.2 will lead to conclusions (1), (2), and (3), respectively of Lemma 6.1, and possibility (4) will lead to conclusion (1).…”

Section: Main Theoremmentioning

confidence: 99%

“…If˛;ˇ; ; ı;˛0;ˇ0; 0 ; ı 0 2 ‫ޑ‬ ‫ޚ‬ then (4) S.˛;ˇI ; ı/ D S.˛0;ˇ0I 0 ; ı 0 / (resp.˙S.˛0;ˇ0I 0 ; ı 0 /) if and only if .˛;ˇ; ; ı/ and .˛0;ˇ0I 0 ; ı 0 / are related by a composition of the transformations in (1) and (2) (resp. the transformations in (1), (2) and (3)). …”

Section: Preliminariesmentioning

confidence: 99%

“…T.K/ may be described as follows; see Boileau and Zimmermann [2]. Let M be the double branched cover of .S 3 ; K/, with covering involution hW M !…”

Section: Preliminariesmentioning

confidence: 99%

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Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

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First we clarify the extension of the main result of [11] to knots in solid tori that is described in the Appendix of [11]. In Section 3, we define a family of hyperbolic knots J " .`; m/ in a solid torus, " 2 f1; 2g and`; m integers, each of which admits a half-integer surgery yielding a toroidal manifold. (The knots J " .`; m/ in the solid torus are the analogs of the knots k.`; m; n; p/ in the 3-sphere.) We then use [11] to show that these are the only such:Theorem 4.2 Let J be a knot in a solid torus whose exterior is irreducible and atoroidal. Let be the meridian of J and suppose that J. / contains an essential torus for some with . ; / 2. Then . ; / D 2 and J D J " .`; m/ for some ";`; m.This theorem along with the main result of [11] then allows us to describe in Theorem 5.2 the relationship between the torus decomposition of the exterior of a knot K in S 3 and the torus decompositon of any non-integral surgery on K . In particular, Theorem 5.2 says that the canonical tori of the exterior of K and of the Dehn surgery will be the same unless K is a cable knot (in which case an essential torus of the knot exterior can become compressible), or K is a k.`; m; n; p/, or K is a satellite with pattern J " .`; m/ (in the latter cases a new essential torus is created).We apply these theorems about non-integral Dehn surgeries to address questions about unknotting number. The knots k.`;m;n;p/ (J " .`; m/) are strongly invertible. Their quotients under the involutions give rise to EM-knots, K.`;m;n;p/ (EM-tangles, A " .`; m/ resp.) which have essential Conway spheres and yet can be unknotted (trivialized, resp.) by a single crossing change. Theorem 6.2 descibes when a knot with an essential Conway sphere or 2-torus can have unknotting number 1. This is naturally stated in the context of the characteristic decomposition of a knot along

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“…This contradiction finishes the proof of Lemma 9.2(1). Lemma 9.2(2) and (3) now follow from Lemma 9.1, (1) and (2). Finally, (4) follows from Lemma 8.1 applied to R.1=.…”

Section: Some Algebraic Tangle Calculationsmentioning

confidence: 85%

Section: Preliminariesmentioning

confidence: 99%

Section: Main Theoremmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…T.K/ may be described as follows; see Boileau and Zimmermann [2]. Let M be the double branched cover of .S 3 ; K/, with covering involution hW M !…”

Section: Preliminariesmentioning

confidence: 99%

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Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

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A Geometric Proof of the Fintushel–Stern Formula

Collin

Saveliev

1999

Advances in Mathematics

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The Fintushel Stern formula asserts that the Casson invariant of a Brieskorn homology sphere 7( p, q, r) equals 1Â8 the signature of its Milnor fiber. We give a geometric proof of this formula, as opposite to computational methods used in the original proof. The formula is also refined to relate equivariant Casson invariants to equivariant signatures. Academic PressKey Words: Milnor fiber; Casson invariant; knot signatures.Let 7( p, q, r) be the link of the singularity of f &1 (0) where f : C 3 Ä C is the polynomial f (x, y, z)=x p + y q +z r , and M( p, q, r) its Milnor fiber. Thus M( p, q, r) is a compact, simply connected, smooth parallelizable 4-manifold with boundary 7( p, q, r). We will assume that p, q, and r are pairwise relatively prime positive integers. Then 7( p, q, r) is an integral homology 3-sphere, and one may consider its Casson invariant, *(7( p, q, r)). In [8], while computing the Floer Homology of 7( p, q, r), Fintushel and Stern noticed the intriguing relation *(7( p, q, r))= where _(M( p, q, r)) stands for the signature of the manifold M( p, q, r).

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A Survey of the Impact of Thurston’s Work on Knot Theory

Sakuma

2020

In the Tradition of Thurston

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The ?-orbifold group of a link

Cited by 77 publications

References 46 publications

Knots with unknotting number 1 and essential Conway spheres

Knots with unknotting number 1 and essential Conway spheres

A Geometric Proof of the Fintushel–Stern Formula

A Survey of the Impact of Thurston’s Work on Knot Theory

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