Tunnel leveling, depth, and bridge numbers
Sangbum Cho, Darryl McCullough · Transactions of the American Mathematical Society · 2010
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 $(1+\sqrt {2})^d$, 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.