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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Constructing knot tunnels using giant steps


Authors: Sangbum Cho and Darryl McCullough
Journal: Proc. Amer. Math. Soc. 138 (2010), 375-384
MSC (2000): Primary 57M25
Published electronically: 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 $ 1$ 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.


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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: http://dx.doi.org/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
Published electronically: September 3, 2009
Additional Notes: The research of both authors was supported in part by NSF grant DMS-0802424
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.