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A sufficient condition is given for adding a 2-handle to a 3- manifold and obtaining a 3-manifold with incompressible boundary. The main result is then used to show that a knot in the 3-sphere admitting a partially unknotted spanning surface has Property R. In (P) it was shown that if J is a simple closed curve in the boundary of a 3- dimensional handlebody H and 3H—J is incompressible in H, then the 3-manifold obtained by adding a 2-handle to H along J has incompressible boundary. The . proof in (P) was algebraic. It used the Freiheitssatz from one-relator group theory and was predicated on the fact that the fundamental group of a 3-dimensional handlebody is a finitely generated free group. In this note the above result is generalized to a larger class of 3-manifolds, which includes 3-dimensional handlebodies. The proof is geometric. The main theorem is applied to give a new class of knots that have Property R. I wish to acknowledge helpful conversations with M. Culler, C. Gordon and M. Starbird about generalizations and improvements to the results of this paper. 1. Main theorem. Let M be a 3-manifold, J a two-sided simple closed curve (s.c.c.) in dM and B a 3-cell with the product structure B = D2 x I. The 3- manifold obtained by adding a 2-handle to M along J is the 3-manifold that is the identification space obtained from the disjoint union of M and B via an identification of the annulus 3D2 X I C dB with an annular neighborhood of J in dM. A simple closed curve K in dM — J is coplanar (in dM) with J if K separates dM — J into two components, one of which is planar. If M is compact and K is coplanar with J, then precisely one of the following possibilities occurs: K bounds a disk in dM — J, K is parallel in dM to J, or K separates dM into two components where one is a once-punctured torus T and J C T does not separate T. A properly embedded planar surface P in M is a pre-disk with respect to J if:

Adding a 2-handle to a 3-manifold: an application to property 𝑅