Geometry & Topology

Levelling an unknotting tunnel

Hiroshi Goda, Martin Scharlemann, and Abigail Thompson

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It is a consequence of theorems of Gordon-Reid [J. Knot Theory Ram. 4 (1995) 389–409] and Thompson [Topology 36 (1997) 505–507] 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 [Bull. Fac. Eng. Takushoku Univ. 3 (1992) 219–225], 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.

Article information

Geom. Topol., Volume 4, Number 1 (2000), 243-275.

Received: 17 January 2000
Accepted: 18 September 2000
First available in Project Euclid: 21 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

tunnel unknotting tunnel bridge position thin position Heegaard splitting


Goda, Hiroshi; Scharlemann, Martin; Thompson, Abigail. Levelling an unknotting tunnel. Geom. Topol. 4 (2000), no. 1, 243--275. doi:10.2140/gt.2000.4.243.

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