Algebraic & Geometric Topology

Cabling sequences of tunnels of torus knots

Sangbum Cho and Darryl McCullough

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Abstract

In previous work, we developed a theory of tunnels of tunnel number 1 knots in S3. It yields a parameterization in which each tunnel is described uniquely by a finite sequence of rational parameters and a finite sequence of 0s and 1s, that together encode a procedure for constructing the knot and tunnel. In this paper we calculate these invariants for all tunnels of torus knots

Article information

Source
Algebr. Geom. Topol., Volume 9, Number 1 (2009), 1-20.

Dates
Received: 5 August 2008
Revised: 22 October 2008
Accepted: 11 December 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796954

Digital Object Identifier
doi:10.2140/agt.2009.9.1

Mathematical Reviews number (MathSciNet)
MR2471129

Zentralblatt MATH identifier
1170.57005

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
knot link tunnel torus knot

Citation

Cho, Sangbum; McCullough, Darryl. Cabling sequences of tunnels of torus knots. Algebr. Geom. Topol. 9 (2009), no. 1, 1--20. doi:10.2140/agt.2009.9.1. https://projecteuclid.org/euclid.agt/1513796954


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References

  • E Akbas, A presentation for the automorphisms of the 3–sphere that preserve a genus two Heegaard splitting, Pacific J. Math. 236 (2008) 201–222
  • M Boileau, M Rost, H Zieschang, On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces, Math. Ann. 279 (1988) 553–581
  • S Cho, Homeomorphisms of the 3–sphere that preserve a Heegaard splitting of genus two, Proc. Amer. Math. Soc. 136 (2008) 1113–1123
  • S Cho, D McCullough, Constructing knot tunnels using giant steps
  • S Cho, D McCullough, The tree of knot tunnels, Geom. Topol. 13 (2009) XXX–YYY
  • S Cho, D McCullough, Tunnel leveling, depth, and bridge numbers
  • S Cho, D McCullough, Software Available at \setbox0\makeatletter\@url http://www.math.ou.edu/~dmccullough {\unhbox0
  • Y Moriah, Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988) 465–481
  • M Scharlemann, Automorphisms of the 3–sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana $(3)$ 10 (2004) 503–514
  • M Scharlemann, A Thompson, Unknotting tunnels and Seifert surfaces, Proc. London Math. Soc. $(3)$ 87 (2003) 523–544