ABSTRACT. Combinatorial sutured manifold theory is used to study the effects of attaching a 2–handle to an essential simple closed curve on a genus two boundary component of a compact, orientable 3–manifold. The main results concern degenerating handle additions to a simple 3– manifold and essential surfaces in the exterior of a knot or link obtained by “boring ” a split link or unknot. (Boring is an operation on knots and two-component links which generalizes rational tangle replacement.) Generalizations and new proofs of several well known theorems from classical knot theory are obtained. These include superadditivity of genus under band connect sum and the fact that unknotting number one knots are prime. 1
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