조상범
2018-04-14T04:37:10Z
2018-04-14T04:37:10Z
2011-01
American Mathematical Society, 2011, 361(1), P.259-280
0002-9947
http://www.ams.org/journals/tran/2011-363-01/S0002-9947-2010-05248-1/home.html
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.
The second author was supported in part by NSF grant DMS-0802424.
en
American Mathematical SOC
UNKNOTTING TUNNELS
HEEGAARD
3-SPHERE
PRESERVE
KNOTS
DECOMPOSITIONS
AUTOMORPHISMS
DISTANCE
GENUS-2
SPACES
TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERS
Article
1
363
10.1090/S0002-9947-2010-05248-1
259-280
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
2011209536
S
COLLEGE OF EDUCATION[S]
DEPARTMENT OF MATHEMATICS EDUCATION
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