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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Tunnel leveling, depth, and bridge numbers


Authors: Sangbum Cho and Darryl McCullough
Journal: Trans. Amer. Math. Soc. 363 (2011), 259-280
MSC (2010): Primary 57M25
Published electronically: August 24, 2010
MathSciNet review: 2719681
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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.


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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/S0002-9947-2010-05248-1
PII: S 0002-9947(2010)05248-1
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
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.