TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERSopen access
- Authors
- Cho, Sang bum; McCullough, Darryl
- Issue Date
- Jan-2011
- Publisher
- AMER MATHEMATICAL SOC
- Citation
- TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.363, no.1, pp.259 - 280
- Indexed
- SCIE
SCOPUS
- Journal Title
- TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
- Volume
- 363
- Number
- 1
- Start Page
- 259
- End Page
- 280
- URI
- https://scholarworks.bwise.kr/hanyang/handle/2021.sw.hanyang/169176
- DOI
- 10.1090/S0002-9947-2010-05248-1
- ISSN
- 0002-9947
- 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 + root 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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