Modifying surfaces in 4–manifolds by twist spinning

In this paper, given a knot K , for any integer m we construct a new surface ΣK(m) from a smoothly embedded surface Σ in a smooth 4–manifold X by performing a surgery on Σ . 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,Σ) after the surgery. One of the main results is that for certain pairs (X,Σ), the smooth type of ΣK(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 CP2 , not isotopic to complex curves, but which are topologically unknotted.