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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.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:
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