Geometry & Topology

Levelling an unknotting tunnel

Hiroshi Goda, Martin Scharlemann, and Abigail Thompson

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883285

Digital Object Identifier
doi:10.2140/gt.2000.4.243

Mathematical Reviews number (MathSciNet)
MR1778174

Zentralblatt MATH identifier
0958.57007

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

Keywords
tunnel unknotting tunnel bridge position thin position Heegaard splitting

Citation

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. https://projecteuclid.org/euclid.gt/1513883285


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References

  • J Berge, Embedding the exteriors of one-tunnel knots and links in the 3–Sphere, unpublished preprint
  • D Gabai, Foliations and the topology of 3–manifolds: III, J. Differential Geom. 26 (1987) 479–536
  • H Goda, M Ozawa, M Teragaito, On tangle decompositions of tunnel number one links, J. Knot Theory and its Rami. 8 (1999) 299–320
  • C Gordon, A Reid, Tangle decompositions of tunnel number one knots and links, J. Knot Theory and its Rami. 4 (1995) 389–409
  • T Kobayashi, Classification of unknotting tunnels for two bridge knots, from: “Proceedings of the 1998 Kirbyfest”, Geometry and Topology Monographs, 2 (1999) 259–290
  • K Morimoto, A note on unknotting tunnels for 2–bridge knots, Bulletin of Faculty of Engineering Takushoku University, 3 (1992) 219–225
  • K Morimoto, Planar surfaces in a handlebody and a theorem of Gordon–Reid, from: “Proc. Knots '96”, (S Suzuki, editor) World Sci. Publ. Co. Singapore (1997) 127–146
  • A Thompson, Thin position and bridge number for knots in the 3–sphere, Topology, 36 (1997) 505–507