> \pJava Excel API v2.6 Ba==h\:#8X@"1Arial1Arial1Arial1Arial + ) , * `DC,,title[*]contributor[author]contributor[advisor]keywords[*]date[issued] publisher citationsidentifier[uri]identifier[doi]abstractrelation[journal]relation[volume]relation[no]relation[page]*TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERSprUNKNOTTING TUNNELS;
HEEGAARD;
3SPHERE;
PRESERVE;
KNOTS;
DECOMPOSITIONS;
AUTOMORPHISMS;
DISTANCE;
GENUS2;
SPACES;201101American Mathematical SOC6American Mathematical Society, 2011, 361(1), P.259280Nhttp://www.ams.org/journals/tran/201136301/S000299472010052481/home.html10.1090/S000299472010052481We 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.1TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY3631259280&HQsj&)KLng8$Fe

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