zbMATH — the first resource for mathematics

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

57M25 Knots and links in the \(3\)-sphere (MSC2010)
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.