  We classify nonnegatively curved simply connected 4-manifolds with circle
symmetry up to equivariant diffeomorphisms. The main problem is rule out
knotted curves in the singular set of the orbit space. As an extension of this
work we classify all knots in S^3 which can be realized as an extremal set with
respect to an inner metric on S^3 which has nonnegative curvature in the
Alexandrov sense.
