Modifying surfaces in 4–manifolds by twist spinning

In this paper, given a knot K , for any integer m we construct a new surface K .m/ from a smoothly embedded surface in a smooth 4-manifold X by performing a surgery on . 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; / after the surgery. One of the main results is that for certain pairs .X; /, the smooth type of 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 P 2 , not isotopic to complex curves, but which are topologically unknotted.