Dehn surgeries and P2-reducible 3-manifolds

This note concerns 3-manifolds M obtained by Dehn surgery on a knot in S3, in particular those which contain embedded projective planes. Generically, in the knot-space S3 − N (k), the intersection of a projective plane P2 in M, and any 2-sphere S2 in S3 pierced by k, is a 1-complex which can be viewed as a graph in either the projective plane or the 2-sphere. C.McA. Gordon and J. Luecke have used similar graphs arising as the intersection of two 2-spheres to prove that a knot in S3 is determined by its complement. With a view to adapting their techniques to show that P3 cannot be obtained by Dehn surgery on a knot in S3, we prove here that a “minimal” P2 in M (i.e., a P2 which minimizes the odd number of intersections with the core of the solid torus added during surgery) gives rise to a graph (in S2), which contains no “generalized” Scharlemann cycles. This obstruction is considered as a step towards showing that P3 cannot be obtained by Dehn surgery along a knot in S3.