Dehn Surgery on Knots

Culler, Marc; Gordon, C. McA.; Luecke, John

doi:10.2307/1971311

Cited by 541 publications

(746 citation statements)

References 13 publications

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“…Denote the two components of @X as @ 1 X , @ 2 X , where @ 1 X is the component along which these fillings are made (corresponding to the boundary of the ambient solid torus of J 1 .`; m/). Because infinitely many fillings of X are hyperbolic, either X is hyperbolic or there is a cable space along @ 1 X (Theorem 2.4.4 of [5]). We assume the latter for contradiction.…”

Section: See Figure 42 (The Boxes Correspond To Vertical Twists)mentioning

confidence: 99%

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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Section: See Figure 42 (The Boxes Correspond To Vertical Twists)mentioning

confidence: 99%

Knots with unknotting number 1 and essential Conway spheres

Gordon

Luecke

2006

Algebr. Geom. Topol.

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“…Our methods also yield a weak version of Corollary 4 in [2]: Theorem 3.2. Let K<=S 3 be an amphichieral knot with A'(/C)#O.…”

Section: Theorem Let K Be a Knot In S 3 Then There Is A Sequence Omentioning

confidence: 89%

“…From the way Casson defines his invariant for knots it is obvious that if K c S 3 is a knot with A'(*0#0 then for all nonzero integers n, TT 1 (M(1/«)) admits an irreducible representation into SU (2). From our viewpoint this is less obvious, but we get the strengthened result: In contrast it should be noted that the (-2,3,7) pretzel knot has l\K)=-\2, and S 3 (18/l) has cyclic fundamental group.…”

Section: Theorem Let K Be a Knot In S 3 Then There Is A Sequence Omentioning

confidence: 99%

Casson's invariant and surgery on knots

Frohman

Long

1992

Proceedings of the Edinburgh Mathematical Society

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“…From [1], at least one of the swallow-follow surfaces obtained from the meridional essential surfaces in Theorem 1 is also essential and of higher genus. Hence, the knots from the theorem are also examples of knots having closed essential surfaces of arbitrarily high genus in their complements.…”

Section: Introductionmentioning

confidence: 99%