Abstract
Suppose a 2-handle is attached to an orientable, irreducible 3-manifold M along a curve α contained in ∂ M, obtaining a 3-manifold M α. Suppose ∂ M is compressible, but ∂ M — α is not. It is proved that if M α is nonsimple, i.e., it contains an essential annulus or torus, then either M is also nonsimple or M α contains an annulus or torus which intersects the 2-handle twice. As an application of the main theorems, all satellite knots in S 3 with tunnel number one are found; they are certain satellites of torus knots.
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