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Tangle decompositions of tunnel number one knots and links. (English) Zbl 0841.57012
A nontrivial knot \(K\) in \(S^3\) is said to have tunnel number one if there exists an arc \(I\) embedded in \(S^3\) with \(I \cap K = \partial I \cap K = \partial I\) such that the complement of \(K \cup I\) is homeomorphic to the interior of a handlebody of genus two. F. H. Norwood [Proc. Am. Math. Soc. 86, 143-147 (1982; Zbl 0506.57004)] proved that a knot with tunnel number one is (1-)prime in the sense that it cannot be expressed as a sum of two essential 1-string tangles. M. Scharlemann [Topology Appl. 18, 235-258 (1984; Zbl 0592.57004)] proved that it is also 2-prime. Here ‘2-prime’ is defined similarly to ‘1-prime’ replacing ‘1-string tangle’ with ‘2-string tangle’. The authors of the paper under review proved that a knot with tunnel number one is \(n\)-prime for any \(n\). (The definition of ‘\(n\)-prime’ might be clear.)
Unlike a knot, a link with tunnel number one may not be prime. In fact Norwood [loc. cit.] pointed out the existence of such a link and K. Morimoto [Topology Appl. 59, No. 1, 59-71 (1994; Zbl 0821.57006)] proved that a non-prime link has tunnel number one if and only if it is a sum of the Hopf link and a two-bridge knot. (See also [A. C. Jones, Composite two-generator links have a Hopf link summand (preprint)], which proves that it contains a Hopf link summand.) The authors also generalize Jones’ work (and so a part of Morimoto’s work). They showed that if a link in \(S^3\) with tunnel number one can be expressed as a sum of two tangles, then it has an \(n\)-string Hopf tangle summand. Here \(n\)-string Hopf tangle consists of \(n\) parallel strings and a circle around them. This paper also studies a knot or link in a 3-manifold and closed surface in their complement.
Reviewer: H.Murakami (Tokyo)

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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