Constructing lens spaces by surgery on knots

Fintushel, Ronald; Stern, Ronald J.

doi:10.1007/bf01161380

Cited by 114 publications

(141 citation statements)

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“…The "Fintushel-Stern" knot [8], also known as the pretzel knot of type (−2, 3, 7), is well-known for several reasons, such as admitting seven exceptional surgeries (see Cameron Gordon's discussion in Problem 1.77 of Kirby's problem list [14]). It is fibred and its Alexander polynomial is L(−t), where L is the Lehmer polynomial…”

Section: Fibrations and Fibred Knotsmentioning

confidence: 99%

Ordered groups, eigenvalues, knots, surgery and L-spaces

Clay¹,

Rolfsen²

2011

Math. Proc. Camb. Phil. Soc.

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Abstract. We establish a necessary condition that an automorphism of a nontrivial finitely generated bi-orderable group can preserve a biordering: at least one of its eigenvalues, suitably defined, must be real and positive. Applications are given to knot theory, spaces which fibre over the circle and to the Heegaard-Floer homology of surgery manifolds. In particular, we show that if a nontrivial fibred knot has bi-orderable knot group, then its Alexander polynomial has a positive real root. This implies that many specific knot groups are not bi-orderable. We also show that if the group of a nontrivial knot is bi-orderable, surgery on the knot cannot produce an L-space, as defined by Ozsváth and Szabó.

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Section: Fibrations and Fibred Knotsmentioning

confidence: 99%

Ordered groups, eigenvalues, knots, surgery and L-spaces

Clay¹,

Rolfsen²

2011

Math. Proc. Camb. Phil. Soc.

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“…The question as to when (and which) lens spaces or connected sums of lens spaces can be obtained by Dehn surgery on an iterated torus knot is considered by Fintushel-Stern in [3]. For lens spaces, the case k = 2 of Theorem 7.5(iii) is Theorem 1 of [3].…”

Section: Lemma 52 Ifkj-o Then T(k)< T(j(k)) Dmentioning

confidence: 99%

“…We examine the case of iterated torus knots in some detail, and in particular describe all the Dehn surgeries on these which yield lens spaces or connected sums of lens spaces. This problem has also been studied by Fintushel-Stern in [3]. However, our point of view provides an alternative to the link calculus approach of [3], and moreover some of the statements in [3] are incorrect.2 In §8 we give the proof of Theorem 1.11, making use of the results of §7.…”

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confidence: 99%

“…This problem has also been studied by Fintushel-Stern in [3]. However, our point of view provides an alternative to the link calculus approach of [3], and moreover some of the statements in [3] are incorrect.2 In §8 we give the proof of Theorem 1.11, making use of the results of §7. § §9 and 10 deal with some other aspects of the question whether (K; 0) can be of the form S] X S2 # M, if K ¥= O.…”

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confidence: 99%

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Dehn Surgery and Satellite Knots

Gordon¹

1983

Transactions of the American Mathematical Society

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Abstract. For certain kinds of 3-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in S3 is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain S3, a fake S3, a fake S3 with nonzero Rohlin invariant, S1 X S1, a fake S] X S2, S] X S2 # M with M nonsimply-connected, or a fake lens space.1. Introduction and statement of results. In the present paper we show that for certain kinds of 3-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot reduces to the corresponding question for simple knots. By Thurston's uniformization theorem for Haken manifolds [27], a knot is simple if and only if it is either hyperbolic (i.e. its complement possesses a complete hyperbolic structure with finite volume), or a torus knot. Since the manifolds obtained by Dehn surgery on the latter can be completely described (see [20], and §7), we essentially have a reduction to the case of hyperbolic knots.If Kis an (unoriented) knot in (oriented) S3, and r = m/n G Q L) {oo}, (m, n) = I, let (K; r) denote the closed, oriented 3-manifold obtained by Dehn surgery of type r on K. (We use the terminology of [21, pp. 258-259]; see §2.) The unknot is denoted by O, so that (O; r) is the lens space L(m, n), if | m \¥= 0,1, or S3 = (O; l/n), or S1 X S2 -(O;0). The manifolds to which our arguments apply are those that are atoroidal, that is, contain no imcompressible torus. Our main result may then be stated as follows. (Throughout, all 3-manifolds are oriented, ~ denotes orientation-preserving homeomorphism, and # connected sum.)

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“…Baily and Rolfsen [2] showed that the lens space L(23,7) could be obtained by -23-surgery on the (ll,2)-cable knot about the trefoil knot; Simon discovered similar examples. Fintushel and Stern [4] constructed infinitely many noniterated torus knots, upon which certain surgery yields lens spaces. More recently, Gordon [6] classified the surgery manifolds of all iterated torus knots, Berge [3] and Gabai [5] have independently constructed an infinite collection of knots in solid tori such that certain surgery on them in the solid torus yield D x S , and therefore yield lens spaces when the knots are considered to be in 5 .…”

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confidence: 99%

Seifert Fibered Surgery Manifolds of Composite Knots

Kalliongis¹,

Tsau²

1990

Proceedings of the American Mathematical Society

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Abstract.A classification is given for the composite knots and the Dehn surgery on these knots which yield Seifert fibered surgery manifolds. We prove that if a knot K is the composition of two torus knots, then some (unique) integral surgery on K yields a Seifert fibered manifold, and conversely if the surgery manifold of a composite knot K is Seifert fibered, then K is the composition of two torus knots and the surgery must be integral surgery, which is uniquely determined.In this paper we classify the composite knots and the Dehn surgeries on these knots which yield Seifert fibered surgery manifolds. In [8] Moser conjectured that surgery on a nontorus knot could not yield a Seifert fibered manifold, in particular, could not yield a lens space. Many counterexamples to Moser's conjecture have been found. Baily and Rolfsen [2] showed that the lens space L(23,7) could be obtained by -23-surgery on the (ll,2)-cable knot about the trefoil knot; Simon discovered similar examples. Fintushel and Stern [4] constructed infinitely many noniterated torus knots, upon which certain surgery yields lens spaces. More recently, Gordon [6] classified the surgery manifolds of all iterated torus knots, Berge [3] and Gabai [5] have independently constructed an infinite collection of knots in solid tori such that certain surgery on them in the solid torus yield D x S , and therefore yield lens spaces when the knots are considered to be in 5 . Since cable knots are prime knots and the surgery manifold of a composite knot contains an incompressible torus (see, for example, [6]), none of the above-mentioned knots is a composite knot. Indeed, no nontrivial surgery on a composite knot may yield a lens space. It is then natural to ask if any such surgery manifold is Seifert fibered. In this paper, we show that if a knot K is the composition of two torus knots, then some (unique) integral surgery on K yields a Seifert fibered manifold, and conversely if a surgery manifold of a composite knot K is Seifert fibered, then the surgery

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Constructing lens spaces by surgery on knots

Cited by 114 publications

References 8 publications

Ordered groups, eigenvalues, knots, surgery and L-spaces

Ordered groups, eigenvalues, knots, surgery and L-spaces

Dehn Surgery and Satellite Knots

Seifert Fibered Surgery Manifolds of Composite Knots

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