Banding, twisted ribbon knots, and producing reducible manifolds via Dehn surgery (Q1113487)

It is shown that if the unknot can be obtained by attaching a band to a composite knot or link, then that knot or link must have a summand of bridge number two. The principal corollaries are that a composite twisted ribbon number one knot has a two-bridge summand, that reducible manifolds arising from Dehn surgery on a strongly invertible knot have lens space summands, and that homology spheres arising from Dehn surgery on a strongly invertible knot are prime.