TUNNEL LEVELING, DEPTH, AND BRIDGE NUMBERS

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.