Levelling an unknotting tunnel.

*(English)*Zbl 0958.57007Summary: 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) |

##### References:

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[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 · doi:10.1142/S0218216595000193 |

[5] | T Kobayashi, Classification of unknotting tunnels for two bridge knots, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 259 · Zbl 0962.57003 · emis:journals/UW/gt/GTMon2/paper14.abs.html |

[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 · doi:10.1016/0040-9383(96)00010-9 |

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