Volume 10, issue 1 (2006)

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Modifying surfaces in 4–manifolds by twist spinning

Hee Jung Kim

Geometry & Topology 10 (2006) 27–56

arXiv: math/0411078


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 P2, not isotopic to complex curves, but which are topologically unknotted.

Twist spinning, Seiberg–Witten invariants, branched covers, ribbon knots
Mathematical Subject Classification 2000
Primary: 57R57
Secondary: 14J80, 57R95
Forward citations
Received: 22 July 2004
Accepted: 2 January 2006
Published: 25 February 2006
Proposed: Ronald Fintushel
Seconded: Peter Ozsváth, Ronald Stern
Hee Jung Kim
Department of Mathematics
McMaster University
Ontario L8S 4K1