The depth of a knot tunnel

TL;DR

This paper explores the depth invariant of tunnel number 1 knots, providing formulas, bounds, and explicit calculations that relate tunnel depth to bridge number growth and knot construction methods.

Contribution

It introduces new recursive formulas and bounds connecting tunnel depth, bridge number, and cabling constructions, and computes slope parameters for specific knot tunnels.

Findings

Number of minimal tunnel move sequences for a given tunnel

Minimum bridge number growth rate as a function of tunnel depth

Maximum bridge number for N cabling constructions is the (N+2)nd Fibonacci number

Abstract

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.