TY - JOUR
AU - 조상범
DA - 2011/01
PY - 2011
UR - http://www.ams.org/journals/tran/2011-363-01/S0002-9947-2010-05248-1/home.html
AB - 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.
PB - American Mathematical SOC
KW - UNKNOTTING TUNNELS
KW - HEEGAARD
KW - 3-SPHERE
KW - PRESERVE
KW - KNOTS
KW - DECOMPOSITIONS
KW - AUTOMORPHISMS
KW - DISTANCE
KW - GENUS-2
KW - SPACES
TI - TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERS
IS - 1
VL - 363
DO - 10.1090/S0002-9947-2010-05248-1
T2 - TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
ER -