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(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.

This is the version published by Geometry & Topology on 25 February 2006

References in corpus (1)

• The Seiberg-Witten invariants and 4-manifolds with essential tori

Cited by in corpus (10)

• Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds
• Embedded surfaces with infinite cyclic knot group
• Knotted surfaces in 4-manifolds and stabilizations
• Smooth surfaces with non-simply-connected complements
• Exotic group actions on simply connected smooth 4-manifolds
• Exotically knotted disks and complex curves
• Null-homologous exotic surfaces in 4-manifolds
• Topologically isotopic and smoothly inequivalent 2-spheres in simply connected 4-manifolds whose complement has a prescribed fundamental group
• Smoothly knotted and topologically unknotted nullhomologous surfaces in 4-manifolds
• Exotic embedded surfaces and involutions from Real Seiberg-Witten theory