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Tunnel number one knots satisfy the Poenaru conjecture. (English) Zbl 0592.57004
Let K be a PL knot with tunnel number one. The author in a clear and concise manner demonstrates that K satisfies the Poenaru conjecture. Also, the author shows that K cannot be written as the join of two prime tangles (i.e., K is doubly prime). In addition, the arguments provide a geometric proof of Norwood’s theorem that tunnel number one knots are prime [see F. H. Norwood, Proc. Am. Math. Soc. 86, 143-147 (1982; Zbl 0506.57004)].
Reviewer: B.Clark

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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##### References:
 [1] S. Bleiler, Knots prime on many strings, (to appear in Trans. Am. Math. Soc.). · Zbl 0545.57001 [2] Clarke, B., The Heegaard genus of manifolds obtained by surgery on links and knots, Intern. jour. math. math. sci., 3, 583-589, (1980) · Zbl 0447.57008 [3] Kirby, R.C.; Lickorish, W.B.R., Prime knots and concordance, Math. proc. camb. phil. soc., 86, 437-441, (1979) · Zbl 0426.57001 [4] Lambert, H., Longitude surgery on genus one knots, Pams, 63, 359-362, (1977) · Zbl 0362.55003 [5] Milnor, J., Infinite cyclic coverings, () · Zbl 0179.52302 [6] Norwood, F.H., Every two generator knot is prime, Proc. amer. math. soc., 86, 143-147, (1982) · Zbl 0506.57004 [7] M. Scharlemann, Outermost forks and a theorem of Jaco (to appear). · Zbl 0589.57011 [8] Scharlemann, M., The schoenflies conjecture is true for genus two imbeddings, Topology, 23, 211-217, (1984) · Zbl 0543.57011 [9] Tsukui, Y., On a prime surface of genus 2 and homeomorphic splitting of 3-sphere, Yok. math. jour., 23, 63-75, (1975) · Zbl 0338.57002
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