Adding 2-handles to sutured manifolds

Combinatorial sutured manifold theory is used to compare the effects of attaching a 2–handle to essential simple closed curves on a genus two boundary component of a compact, orientable 3–manifold. The main theorems are applied to both arbitrary 2–handle attachment and to a limited form of 2–handle attachment known as “refilling meridians”. Generalizations and new proofs of several well-known theorems from classical knot theory are obtained, including superadditivity of knot genus under band connect sum and the fact that unknotting number one knots are prime.

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