Modifying surfaces in 4–manifolds by twist spinning

Kim, Hee Jung

doi:10.2140/gt.2006.10.27

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Modifying surfaces in 4–manifolds by twist spinning

Hee Jung Kim

Geometry & Topology 10 (2006) 27–56

DOI: 10.2140/gt.2006.10.27

arXiv: math/0411078

Abstract

In this paper, given a knot $K$ , for any integer $m$ we construct a new surface $Σ_{K} (m)$ 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 $Σ_{K} (m)$ 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 $ℂ P^{2}$ , not isotopic to complex curves, but which are topologically unknotted.

Keywords

Twist spinning, Seiberg–Witten invariants, branched covers, ribbon knots

Mathematical Subject Classification 2000

Primary: 57R57

Secondary: 14J80, 57R95

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Publication

Received: 22 July 2004

Accepted: 2 January 2006

Published: 25 February 2006

Proposed: Ronald Fintushel

Seconded: Peter Ozsváth, Ronald Stern

Authors

Hee Jung Kim
Department of Mathematics McMaster University Hamilton Ontario L8S 4K1 Canada

Volume 10, issue 1 (2006)