2000
DOI: 10.1007/s000140050131
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Additivity of tunnel number for small knots

Abstract: Abstract. We show that for small knots

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Cited by 55 publications
(34 citation statements)
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“…A strongly irreducible Heegaard splitting can be isotoped so that its splitting surface, S, intersects an incompressible surface, P, only in curves essential in both S and P. This is a deep fact and is proven, for instance, in [14,Lemma 6]. This fact, together with the fact that incompressible surfaces can be isotoped to meet only in essential curves, establishes the following:…”
Section: Definition 14mentioning
confidence: 65%
“…A strongly irreducible Heegaard splitting can be isotoped so that its splitting surface, S, intersects an incompressible surface, P, only in curves essential in both S and P. This is a deep fact and is proven, for instance, in [14,Lemma 6]. This fact, together with the fact that incompressible surfaces can be isotoped to meet only in essential curves, establishes the following:…”
Section: Definition 14mentioning
confidence: 65%
“…See [14]. Now we give an extension of Schultens's lemma as follow: Proof By Haken's lemma, M is irreducible.…”
Section: Definition 22mentioning
confidence: 99%
“…Since c > 0, X .c/ admits an essential torus T that gives the decomposition X .c/ D X 0 [ T Q .c/ , where X 0 Š X and Q .c/ is a c -times punctured annulus cross S 1 . Since T is incompressible and † is strongly irreducible, we may isotope † so that every component of † \ T is essential in both surfaces (see, for example, Schultens [26,Lemma 6]). Isotope † to minimize j † \ T j subject to this constraint.…”
Section: Decomposing Xmentioning
confidence: 99%