Levelling an unknotting tunnel. (English) Zbl 0958.57007

Summary: It is a consequence of theorems of C. McA. Gordon and A. W. Reid [J. Knot Theory Ramifications 4, No. 3, 389-409 (1995; Zbl 0841.57012)] and A. Thompson [Topology 36, No. 2, 505-507 (1997; 867.57009)] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [A note on unknotting tunnels for 2-bridge knots, Bulletin of Faculty of Engineering Takushoku University 3, 219-225 (1992)], who showed that the (now known) classification of unknotting tunnels for 2-bridge knots would follow quickly if it were known that any unknotting tunnel can be made level.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
Full Text: DOI arXiv EuDML EMIS


[1] J Berge, Embedding the exteriors of one-tunnel knots and links in the 3-Sphere, unpublished preprint
[2] D Gabai, Foliations and the topology of 3-manifolds III, J. Differential Geom. 26 (1987) 479 · Zbl 0639.57008
[3] H Goda, M Ozawa, M Teragaito, On tangle decompositions of tunnel number one links, J. Knot Theory Ramifications 8 (1999) 299 · Zbl 0941.57011
[4] C M Gordon, A W Reid, Tangle decompositions of tunnel number one knots and links, J. Knot Theory Ramifications 4 (1995) 389 · Zbl 0841.57012
[5] T Kobayashi, Classification of unknotting tunnels for two bridge knots, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 259 · Zbl 0962.57003
[6] K Morimoto, A note on unknotting tunnels for 2-bridge knots, Bull. Fac. Eng. Takushoku Univ. 3 (1992) 219
[7] K Morimoto, Planar surfaces in a handlebody and a theorem of Gordon-Reid, World Sci. Publ., River Edge, NJ (1997) 123 · Zbl 0969.57016
[8] A Thompson, Thin position and bridge number for knots in the 3-sphere, Topology 36 (1997) 505 · Zbl 0867.57009
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.