## Geometry & Topology

### Modifying surfaces in 4–manifolds by twist spinning

Hee Jung Kim

#### Abstract

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.

#### Article information

Source
Geom. Topol., Volume 10, Number 1 (2006), 27-56.

Dates
Accepted: 2 January 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799702

Digital Object Identifier
doi:10.2140/gt.2006.10.27

Mathematical Reviews number (MathSciNet)
MR2207789

Zentralblatt MATH identifier
1104.57018

#### Citation

Kim, Hee Jung. Modifying surfaces in 4–manifolds by twist spinning. Geom. Topol. 10 (2006), no. 1, 27--56. doi:10.2140/gt.2006.10.27. https://projecteuclid.org/euclid.gt/1513799702

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