Levelling an unknotting tunnel

Abstract

It is a consequence of theorems of Gordon-Reid [4] and Thompson [8] that a tunnel number one knot, if put in thin position, will also be in bridge position. We show that in such a thin presentation, the tunnel can be made level so that it lies in a level sphere. This settles a question raised by Morimoto [6], who showed that the (now known) classification of unknotting tunnels for 2-bridge knots would follow quickly if it were known that any unknotting tunnel can be made level.