The depth of a knot tunnel
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The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it equals the minimum number of Goda-Scharlemann-Thompson tunnel moves 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 rate of growth of this value improves the known estimates of the 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)nd Fibonacci number. Finally, we explicitly compute the slope parameters for the "short" tunnels of torus knots. In particular, we find a sequence of such tunnels for which the bridge numbers of the associated knots, as a function of the depth, achieve the minimum growth rate. The actual minimum bridge number at a given depth cannot be achieved by a torus knot.