Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

Constructing knot tunnels using giant steps

Author(s): Sangbum Cho; Darryl McCullough
Journal: Proc. Amer. Math. Soc. 138 (2010), 375-384.
MSC (2000): Primary 57M25
Posted: September 3, 2009
MathSciNet review: 2550203
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Abstract: In 2000, Goda, Scharlemann, and Thompson described a general construction of all tunnels of tunnel number knots using ``tunnel moves''. The theory of tunnels introduced by Cho and McCullough provides a combinatorial approach to understanding tunnel moves. We use it to calculate the number of distinct minimal sequences of such moves that can produce a given tunnel. As a consequence, we see that for a sparse infinite set of tunnels, the minimal sequence is unique, but generically a tunnel will have many such constructions. Finally, we give a characterization of the tunnels with a unique minimal sequence.

References:

1.: E. Akbas, A presentation of the automorphisms of the -sphere that preserve a genus two Heegaard splitting, Pacific J. Math. 236 (2008), 201-222. MR 2407105 (2009d:57029)
2.: S. Cho, Homeomorphisms of the -sphere that preserve a Heegaard splitting of genus two, Proc. Amer. Math. Soc. 136 (2008), 1113-1123. MR 2361888 (2009c:57029)
3.: S. Cho and D. McCullough, The tree of knot tunnels, Geom. Topol. 13 (2009), 769-815. MR 2469530
4.: S. Cho and D. McCullough, Tunnel leveling, depth, and bridge numbers, arXiv:0812.1396.
5.: S. Cho and D. McCullough, software available at www.math.ou.edu/ dmccullough/research/ software.html .
6.: H. Goda, M. Scharlemann, and A. Thompson, Levelling an unknotting tunnel, Geom. Topol. 4 (2000), 243-275. MR 1778174 (2002h:57011)
7.: M. Scharlemann, Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana (3) 10 (2004), 503-514. MR 2199366 (2007c:57020)
8.: M. Scharlemann and A. Thompson, Unknotting tunnels and Seifert surfaces, Proc. London Math. Soc. (3) 87 (2003), 523-544. MR 1990938 (2004e:57015)

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Additional Information:

Sangbum Cho
Affiliation: Department of Mathematics, University of California at Riverside, Riverside, California 92521
Email: scho@math.ucr.edu

Darryl McCullough
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: dmccullough@math.ou.edu

DOI: 10.1090/S0002-9939-09-10069-2
PII: S 0002-9939(09)10069-2
Received by editor(s): July 29, 2008,
Received by editor(s) in revised form: May 8, 2009
Posted: September 3, 2009
Additional Notes: The research of both authors was supported in part by NSF grant DMS-0802424
Communicated by: Daniel Ruberman
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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