Dehn surgery on a knot with three bridges cannot yield \(P^ 3\) (Q1355880)

The main result of the paper is the statement of the title. Equivalently, the complement of a nontrivial 3-bridge knot in the 3-sphere \(S^3\) cannot be homeomorphic to the complement of a knot in projective 3-space \(P^3\). This has been known previously for several classes of knots as torus knots, satellite knots, symmetric knots, knots of genus one and 2-bridge knots. Gordon and Luecke proved that the 3-sphere cannot be obtained by nontrivial Dehn surgery on a nontrivial knot in \(S^3\) (``knots are determined by their complement''). The present paper follows their approach, using also the concept of ``thin position'' developed by Gabai in his proof of Property \(R\) for knots (surgery on a nontrivial knot in \(S^3\) does not give \(S^2 \times S^1)\). So the main point of the proof is a careful combinatorial and graph-theoretical analysis of the intersection pattern of two surfaces in the exterior of the knot (the 3-bridge condition imposes some restriction on the number of boundary components of one of these surfaces).