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

DOI:
https://doi.org/10.1090/S0002-9947-2010-05248-1

Published electronically:
August 24, 2010

MathSciNet review:
2719681

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Abstract: We use the theory of tunnel number 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 knots. In particular, we calculate the minimum bridge number of a knot as a function of the maximum depth invariant of its tunnels. The growth of this value is on the order of , 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 cabling constructions is the 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:
https://doi.org/10.1090/S0002-9947-2010-05248-1

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.