Modifying surfaces in 4–manifolds by twist spinning

Abstract

In this paper, given a knot $K$, for any integer $m$ we construct a new surface ${\Sigma }_K\left(m\right)$ 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 $\left(X,\Sigma \right)$ after the surgery. One of the main results is that for certain pairs $\left(X,\Sigma \right)$, the smooth type of ${\Sigma }_K\left(m\right)$ 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 $ℂ{\mathbf{P}}^2$, not isotopic to complex curves, but which are topologically unknotted.