Geodesic
On a Realization of Prime Tangles and Knots
Canadian journal of mathematics, Tome 35 (1983) no. 2, pp. 311-323

Voir la notice de l'article provenant de la source Cambridge University Press

The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles. 
On a Realization of Prime Tangles and Knots. Canadian journal of mathematics, Tome 35 (1983) no. 2, pp. 311-323. doi: 10.4153/CJM-1983-017-8
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