Tunnel leveling, depth, and bridge numbers
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- by Sangbum Cho and Darryl McCullough
- Trans. Amer. Math. Soc. 363 (2011), 259-280
- DOI: https://doi.org/10.1090/S0002-9947-2010-05248-1
- Published electronically: August 24, 2010
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Abstract:
We use the theory of tunnel number $1$ knots introduced in an earlier paper to strengthen the Tunnel Leveling Theorem of Goda, Scharlemann, and Thompson. This yields considerable information about bridge numbers of tunnel number $1$ knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant $d$ of its tunnels. The growth of this value is on the order of $(1+\sqrt {2})^d$, which improves known estimates of the rate of growth of bridge number as a function of the Hempel distance of the associated Heegaard splitting. We also find the maximum bridge number as a function of the number of cabling constructions needed to produce the tunnel, showing in particular that the maximum bridge number of a knot produced by $n$ cabling constructions is the $(n+2)^{nd}$ Fibonacci number. Finally, we examine the special case of the “middle” tunnels of torus knots.References
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Bibliographic 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): January 21, 2009
- Published electronically: August 24, 2010
- Additional Notes: The second author was supported in part by NSF grant DMS-0802424
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 259-280
- MSC (2010): Primary 57M25
- DOI: https://doi.org/10.1090/S0002-9947-2010-05248-1
- MathSciNet review: 2719681