Abstract: A tame knot K in the 3-sphere is said to have Property P if no non-trivial surgery along K yields a homotopy 3-sphere. It was conjectured in the early 1970's that every non-trivial knot has Property P. The conjecture is known to hold for a large class of knots, a class that includes non-trivial torus knots, satellite knots, and alternating knots. If it holds in general, then surgery on a knot cannot produce a counterexample to the Poincare Conjecture. We study the Property P Conjecture via knot complement cone complexes (kccc's). These are 3-complexes which consist of a knot complement in a cube with handles, together with the cone over the outer boundary component. Given a knot K and a prescribed surgery, we produce a kccc which is (up to 3-deformation) a spine of the surgered manifold. Thus, kccc's account for all homotopy 3-spheres produced by a surgery on a knot. We hope, with this new view, to get information about these homotopy 3-spheres. In particular we ask: Must every such homotopy 3-sphere have a spine which 3-deforms to a point?