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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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