Geodesic
Levelling an unknotting tunnel
Goda, Hiroshi1 ; Scharlemann, Martin2 ; Thompson, Abigail3
1 Graduate School of Science and Technology, Kobe University, Rokko, Kobe 657-8501, Japan
2 Mathematics Department, University of California, Santa Barbara, California 93106, USA
3 Mathematics Department, University of California, Davis, CA 95616, USA
Geometry & topology, Tome 4 (2000) no. 1, pp. 243-275.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

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.

DOI : 10.2140/gt.2000.4.243
Keywords: tunnel, unknotting tunnel, bridge position, thin position, Heegaard splitting

Goda, Hiroshi 1 ; Scharlemann, Martin 2 ; Thompson, Abigail 3

1 Graduate School of Science and Technology, Kobe University, Rokko, Kobe 657-8501, Japan
2 Mathematics Department, University of California, Santa Barbara, California 93106, USA
3 Mathematics Department, University of California, Davis, CA 95616, USA 
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Goda, Hiroshi; Scharlemann, Martin; Thompson, Abigail. Levelling an unknotting tunnel. Geometry & topology, Tome 4 (2000) no. 1, pp. 243-275. doi : 10.2140/gt.2000.4.243. http://geodesic.mathdoc.fr/articles/10.2140/gt.2000.4.243/

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[3] H Goda, M Ozawa, M Teragaito, On tangle decompositions of tunnel number one links, J. Knot Theory Ramifications 8 (1999) 299

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[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, from: "KNOTS '96 (Tokyo)", World Sci. Publ., River Edge, NJ (1997) 123

[8] A Thompson, Thin position and bridge number for knots in the 3–sphere, Topology 36 (1997) 505

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