The Depth of a Knot Tunnel

The theory of tunnel number 1 knots detailed in [4] provides a non-negative integer invariant depth(τ ) for a knot tunnel τ . We give various results related to the depth invariant. Noting that it equals the minimal number of Goda-Scharlemann-Thompson “tunnel moves” [6] needed to construct the tunnel, we calculate the number of distinct minimal sequences of tunnel moves that can produce a given tunnel. Next, we give a recursion that tells the minimum bridge number of a knot having a tunnel of depth d. The growth of this value is proportional to (1 + √ 2), 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 give various upper bounds for bridge number in terms of the cabling constructions needed to produce a tunnel of a knot, showing in particular that the maximum bridge number of a knot produced by n cabling constructions is the (n + 2) Fibonacci number. Finally, we explicitly compute the slope parameters for the regular (or “short”) tunnels of torus knots, and find a sequence of them for which the bridge numbers of the associated knots achieve the growth rate (1 + √