THE PROJECTIONS OF n-KNOTS WHICH ARE NOT THE PROJECTION OF ANY UNKNOTTED KNOT

Let n be any integer greater than two. We prove that there exists a projection P having the following properties. (1) P is not the projection of any unknotted knot. (2) The singular point set of P consists of double points. (3) P is the projection of an n-knot which is diffeomorphic to the standard sphere.

We prove there exists an immersed n-sphere (⊂ℝⁿ⁺¹×{0}) which is not the projection of any n-knot (n>2). Note that the second theorem is different from the first one.