TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERS

Title
TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERS
Author
조상범
Keywords
UNKNOTTING TUNNELS; HEEGAARD; 3-SPHERE; PRESERVE; KNOTS; DECOMPOSITIONS; AUTOMORPHISMS; DISTANCE; GENUS-2; SPACES
Issue Date
2011-01
Publisher
American Mathematical SOC
Citation
American Mathematical Society, 2011, 361(1), P.259-280
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.
URI
http://www.ams.org/journals/tran/2011-363-01/S0002-9947-2010-05248-1/home.html
ISSN
0002-9947
DOI
10.1090/S0002-9947-2010-05248-1
Appears in Collections:
COLLEGE OF EDUCATION[S](사범대학) > MATHEMATICS EDUCATION(수학교육과) > Articles
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