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Knots prime on many strings. (English) Zbl 0545.57001
A tangle (\({\mathcal B},t)\) means a 3-ball \({\mathcal B}\) with a finite number of properly imbedded disjoint spanning arcs \(t\). A tangle (\({\mathcal B},t)\) is said to be prime if the following three properties are satisfied: (1) Any 2-sphere imbedded in \({\mathcal B}\) which meets the strings transversely in two points bounds a 3-ball in \({\mathcal B}\) which meets \(t\) in a single unknotted arc; (2) Any properly imbedded disc \({\mathcal D}\) which meets \(t\) transversely in a single point is such that \(\delta {\mathcal D}\) bounds a disc in \(\delta {\mathcal B}\) which also meets the strings in the single point; (3) No properly imbedded disc separates the strings. A prime knot \({\mathcal K}\) in the 3-sphere is said to be prime on \(n\)-strings if there is no imbedded 2-sphere intersecting the knot transversely which separates (\({\mathcal S}^ 3,{\mathcal K})\) into prime \(n\)-strings tangles. A knot which is prime on two strings is said to be doubly prime. The author gives many examples of prime knots on \(n\)-strings and a new characterization of knot primality as Theorem 3.1: A knot \({\mathcal K}\) in the 3-sphere is prime if and only if any general double of \({\mathcal K}\) is double prime.
Reviewer: Y.Nakanishi

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M12 Low-dimensional topology of special (e.g., branched) coverings
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