Incompressibility of surfaces in surgered 3-manifolds

Wu, Ying-Qing

doi:10.1016/0040-9383(92)90020-i

Cited by 81 publications

(61 citation statements)

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“…In general this bound is not best possible, and in some cases stronger results are known, but as it is sufficient for Theorems 1.2, 1.3, and 1.4 we do not discuss the matter further. (For some recent results on these questions see [Wu1], [GLu], [BZ1], [BZ2], [H] and [HM]. )…”

Section: Theorem 14 Suppose That M ∈ H and That ∂M Is A Torus Suppmentioning

confidence: 99%

Boundary slopes of punctured tori in 3-manifolds

Gordon

1998

Trans. Amer. Math. Soc.

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Abstract. Let M be an irreducible 3-manifold with a torus boundary component T , and suppose that r, s are the boundary slopes on T of essential punctured tori in M , with their boundaries on T . We show that the intersection number ∆(r, s) of r and s is at most 8. Moreover, apart from exactly four explicit manifolds M , which contain pairs of essential punctured tori realizing ∆(r, s) = 8, 8, 7 and 6 respectively, we have ∆(r, s) ≤ 5. It follows immediately that if M is atoroidal, while the manifolds M (r), M(s) obtained by rand s-Dehn filling on M are toroidal, then ∆(r, s) ≤ 8, and ∆(r, s) ≤ 5 unless M is one of the four examples mentioned above.Let H 0 be the class of 3-manifolds M such that M is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that M ∈ H 0 , where ∂M has a torus component T , and ∂M − T = ∅. Let r, s be slopes on T such that M (r), M(s) / ∈ H 0 . Then ∆(r, s) ≤ 5. The exterior of the Whitehead sister link shows that this bound is best possible.

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Section: Theorem 14 Suppose That M ∈ H and That ∂M Is A Torus Suppmentioning

confidence: 99%

Boundary slopes of punctured tori in 3-manifolds

Gordon

1998

Trans. Amer. Math. Soc.

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“…The topological properties of (1, 1)-knots, also called genus one 1-bridge knots, have recently been investigated in several papers (see [1,5,6,8,9,10,12,13,14,15,18,19,20,21,24,25,26]). These knots are very important in the light of some results and conjectures involving Dehn surgery on knots (see in particular [9] and [25]).…”

Section: Introductionmentioning

confidence: 99%

(1, 1)-knots via the mapping class group of the twice punctured torus

2004

Advg

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We develop an algebraic representation for (1, 1)-knots using the mapping class group of the twice punctured torus M CG 2 (T ). We prove that every (1, 1)-knot in a lens space L(p, q) can be represented by the composition of an element of a certain rank two free subgroup of M CG 2 (T ) with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k, ck + 2).

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“…Suppose M is an irreducible 3-manifold with torus T as a boundary component, and suppose P is an incompressible surface on dM -T. If P is compressible in M(yx) and if M(y2) is reducible, then A(yx, y2) < I. Proposition 2 [9]. Let M be a 3-manifold with torus T as a boundary component, and let P be an incompressible surface in dM -T. Suppose there is no incompressible annulus with one boundary component in P and the other in T. If P is compressible in both M(yx) and M(y2), then A(yx, y2) < 1.…”

Section: Introductionmentioning

confidence: 99%

“…Suppose AZ is a 3-manifold with torus F asa boundary component, and let P be an incompressible surface on <9AZ disjoint from T. It was proved in [9] that in most cases, P remains incompressible in most of the Dehn filled manifolds M(y). (See Proposition 2 below.)…”

Section: Introductionmentioning

confidence: 99%

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Essential laminations in surgered 3-manifolds

Wu¹

1992

Proc. Amer. Math. Soc.

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Abstract.In generic cases, an essential lamination in the interior of a 3-manifold will remain essential after most of the Dehn fillings along a torus boundary component.Suppose AZ is a 3-manifold with torus F asa boundary component, and let P be an incompressible surface on <9AZ disjoint from T. It was proved in [9] that in most cases, P remains incompressible in most of the Dehn filled manifolds M(y). (See Proposition 2 below.) The present note is to solve a problem posed informally by Peter Shalen, which asks whether a similar result holds for essential laminations in 3-manifolds. The answer is positive: Essential laminations disjoint from T will usually remain essential after Dehn fillings. Note that the essentiality of a lamination concerns not only compressibility but also reducibility and existence of end compressing disks in the resulting manifolds.Throughout this paper, all 3-manifolds are assumed orientable. Let AZ be a 3-manifold and T a torus component of the boundary of AZ. For a slope y on T, let M(y) be the manifold obtained by attaching a solid torus V to T such that y corresponds to the meridian slope of V. If yx, y2 are two slopes on T, denote their geometric intersection number by A = A(j>i, y2). We refer the readers to [4] for the definition of essential laminations and related notions like branched surfaces, horizontal boundary, vertical boundary, monogons, etc. Now suppose M is compact, and let X be an essential lamination in AZ. We assume that X is disjoint from ÖAZ. A set A is called a quasi-annulus if A is the image of a continuous map /: Sx x I -► AZ, such that f\s¡ X[o, i) is an embedding, f(Sx x 0) = A n T, and f(Sx x 1) = A n X. A is incompressible if f(Sx x [0, 1)) is incompressible. Theorem 1. Suppose M contains no incompressible quasi-annulus from T to X. If X is not essential in M(yx) and M(y2), then A(yx, y2) < 1. Thus X is essential in M(y) for all but at most three slopes y. Remark. Generally, the quasi-annulus in the theorem cannot be replaced by an embedded annulus. For example, let A be a Klein bottle, and let Mx be a twisted I-bundle over X. Let AZ2 be a (p, q) cable space, q > 1. (See e.g.,

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Incompressibility of surfaces in surgered 3-manifolds

Cited by 81 publications

References 4 publications

Boundary slopes of punctured tori in 3-manifolds

Boundary slopes of punctured tori in 3-manifolds

(1, 1)-knots via the mapping class group of the twice punctured torus

Essential laminations in surgered 3-manifolds

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