  In this paper, given a knot K, for any integer m we construct a new surface
Sigma_K(m) from a smoothly embedded surface Sigma in a smooth 4-manifold X by
performing a surgery on Sigma. This surgery is based on a modification of the
`rim surgery' which was introduced by Fintushel and Stern, by doing additional
twist spinning. We investigate the diffeomorphism type and the homeomorphism
type of (X,Sigma) after the surgery. One of the main results is that for
certain pairs (X,Sigma), the smooth type of Sigma_K(m) can be easily
distinguished by the Alexander polynomial of the knot K and the homeomorphism
type depends on the number of twist and the knot. In particular, we get new
examples of knotted surfaces in CP^2, not isotopic to complex curves, but which
are topologically unknotted.
