#### Geometry & Topology, Vol. 4 (2000)
Paper no. 8, pages 243--275.

## Levelling an unknotting tunnel

### Hiroshi Goda, Martin Scharlemann, Abigail Thompson

**Abstract**.
It is a consequence of theorems of Gordon-Reid [Tangle decompositions
of tunnel number one knots and links, J. Knot Theory and its
Ramifications, 4 (1995) 389-409] and Thompson [Thin position and
bridge number for knots in the 3-sphere, 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 [A note on unknotting tunnels
for 2-bridge knots, Bulletin of Faculty of Engineering Takushoku
University, 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.
**Keywords**.
Tunnel, unknotting tunnel, bridge position, thin position, Heegaard splitting

**AMS subject classification**.
Primary: 57M25.
Secondary: 57M27.

**DOI:** 10.2140/gt.2000.4.243

**E-print:** `arXiv:math.GT/9910099`

Submitted to GT on 17 January 2000.
Paper accepted 18 September 2000.
Paper published 3 October 2000.

Notes on file formats
Hiroshi Goda, Martin Scharlemann, Abigail Thompson

Graduate School of Science and Technology, Kobe University

Rokko, Kobe 657-8501, Japan

Mathematics Department, University of California

Santa Barbara, CA 93106, USA

Mathematics Department, University of California

Davis, CA 95616, USA

Email: goda@math.kobe-u.ac.jp, mgscharl@math.ucsb.edu, thompson@math.ucdavis.edu

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