In this paper, given a knot , for any integer we construct a new surface from a smoothly embedded surface in a smooth 4–manifold 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 after the surgery. One of the main results is that for certain pairs , the smooth type of can be easily distinguished by the Alexander polynomial of the knot and the homeomorphism type depends on the number of twist and the knot. In particular, we get new examples of knotted surfaces in , not isotopic to complex curves, but which are topologically unknotted.
"Modifying surfaces in 4–manifolds by twist spinning." Geom. Topol. 10 (1) 27 - 56, 2006. https://doi.org/10.2140/gt.2006.10.27