In [Scharlemann 1985a] a complicated combinatorial argument showed that the band sum of knots is unknotted if and only if the band sum is a connected sum of unknots. This argument has since been dramatically simplified [Thompson 1987] and extended [Gabai 1987; S3, Sect. 8] using the newly developed machinery of Gabai. In [Scharlemann 1985b] a similar but more complicated combinatorial argument demonstrated that unknotting number one knots are prime. It seems natural to ask whether the Gabai machinery can simplify the proof of this old conjecture as well.
In fact the Gabai machine reveals a connection between the unknotting number of a knot, its genus, and the position of its companion tori. In Sect. 3 we show (roughly) that, if a single crossing change made to a knot \( K \) reduces its genus by more than one, then any companion torus to \( K \) can be made disjoint from the crossing. In particular, of \( K \) were composite and of unknotting number one, then the swallow-follow companion torus would remain as a companion to the unknot, which is impossible. Therefore no composite knot has unknotting number one.
This argument exploits the drop (by at least two) in the genus of a composite knot when a crossing change unknots it. It is natural to ask whether, in general, the genus of a knot drops (or at least does not rise) as it is unknotted by crossing changes. Knots exist for which a crossing change both lowers the unknotting number and raises the genus. A specific example (due to Chuck Livingston) is given in the appendix. Boileau and Murakami have shown us others. In Sect. 1 we give a general construction, again using the Gabai machine, which seems to produce myriads of examples.
Section 2 is a technical section which readies the Gabai machine (as presented in [Scharlemann 1989]) for use in Sect. 3. In Sect. 4 we view crossing changes as a special case of a more general operation, that of attaching an \( n \)-half-twisted band, and discuss how the main results of Sect. 3 generalize.
