Algebraic & Geometric Topology

An algorithm for finding parameters of tunnels

Kai Ishihara

Abstract

Cho and McCullough gave a numerical parameterization of the collection of all tunnels of all tunnel number 1 knots and links in the $3$–sphere. Here we give an algorithm for finding the parameter of a given tunnel by using its Heegaard diagram.

Article information

Source
Algebr. Geom. Topol., Volume 11, Number 4 (2011), 2167-2190.

Dates
Revised: 17 May 2011
Accepted: 30 May 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715265

Digital Object Identifier
doi:10.2140/agt.2011.11.2167

Mathematical Reviews number (MathSciNet)
MR2826935

Zentralblatt MATH identifier
1232.57010

Citation

Ishihara, Kai. An algorithm for finding parameters of tunnels. Algebr. Geom. Topol. 11 (2011), no. 4, 2167--2190. doi:10.2140/agt.2011.11.2167. https://projecteuclid.org/euclid.agt/1513715265

References

• J Berge, Embedding the exteriors of one-tunnel knots and links in the $3$–sphere, unpublished manuscript
• S Cho, D McCullough, Cabling sequences of tunnels of torus knots, Algebr. Geom. Topol. 9 (2009) 1–20
• S Cho, D McCullough, The tree of knot tunnels, Geom. Topol. 13 (2009) 769–815
• S Cho, D McCullough, Tunnel leveling, depth, and bridge numbers, Trans. Amer. Math. Soc. 363 (2011) 259–280
• A T Fomenko, S V Matveev, Algorithmic and computer methods for three-manifolds, Math. and its Appl. 425, Kluwer, Dordrecht (1997) Translated from the 1991 Russian original by M Tsaplina and M Hazewinkel and revised by the authors, With a preface by Hazewinkel
• H Goda, C Hayashi, Genus two Heegaard splittings of exteriors of $1$–genus $1$–bridge knots, preliminary version
• H Goda, C Hayashi, Genus two Heegaard splittings of exteriors of $1$–genus $1$–bridge knots
• C M Gordon, J Luecke, Knots are determined by their complements, Bull. Amer. Math. Soc. $($N.S.$)$ 20 (1989) 83–87
• T Homma, M Ochiai, M-o Takahashi, An algorithm for recognizing $S\sp{3}$ in $3$–manifolds with Heegaard splittings of genus two, Osaka J. Math. 17 (1980) 625–648
• K Morimoto, M Sakuma, On unknotting tunnels for knots, Math. Ann. 289 (1991) 143–167
• M Ochiai, Heegaard diagrams and Whitehead graphs, Math. Sem. Notes Kobe Univ. 7 (1979) 573–591
• M Scharlemann, A Thompson, Unknotting tunnels and Seifert surfaces, Proc. London Math. Soc. $(3)$ 87 (2003) 523–544