  We prove that knots obtained by attaching a band to a split link satisfy the
cabling conjecture. We also give new proofs that unknotting number one knots
are prime and that genus is superadditive under band sum. Additionally, we
prove a collection of results comparing two 2-handle additions to a genus two
boundary component of a compact, orientable 3-manifold. These results give a
near complete solution to a conjecture of Scharlemann and provide evidence for
a conjecture of Scharlemann and Wu. The proofs make use of a new theorem
concerning the effects of attaching a 2-handle to a suture in the boundary of a
sutured manifold.
