Constructing knot tunnels using giant steps
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- by Sangbum Cho and Darryl McCullough PDF
- Proc. Amer. Math. Soc. 138 (2010), 375-384 Request permission
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.References
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Additional Information
- Sangbum Cho
- Affiliation: Department of Mathematics, University of California at Riverside, Riverside, California 92521
- MR Author ID: 830719
- Email: scho@math.ucr.edu
- Darryl McCullough
- Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
- Email: dmccullough@math.ou.edu
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 375-384
- MSC (2000): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9939-09-10069-2
- MathSciNet review: 2550203