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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Tunnel leveling, depth, and bridge numbers
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by Sangbum Cho and Darryl McCullough PDF
Trans. Amer. Math. Soc. 363 (2011), 259-280 Request permission

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
  • 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