Volume 4, issue 1 (2000)

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Levelling an unknotting tunnel

Hiroshi Goda, Martin Scharlemann and Abigail Thompson

Geometry & Topology 4 (2000) 243–275

arXiv: math.GT/9910099

Abstract

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.

Keywords
tunnel, unknotting tunnel, bridge position, thin position, Heegaard splitting
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27
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Publication
Received: 17 January 2000
Accepted: 18 September 2000
Published: 3 October 2000
Proposed: Cameron Gordon
Seconded: Robion Kirby, David Gabai
Authors
Hiroshi Goda
Graduate School of Science and Technology
Kobe University
Rokko
Kobe 657-8501
Japan
Martin Scharlemann
Mathematics Department
University of California
Santa Barbara
California 93106
USA
Abigail Thompson
Mathematics Department
University of California
Davis
CA 95616
USA