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.

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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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