Geometry & Topology

Modifying surfaces in 4–manifolds by twist spinning

Hee Jung Kim

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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.

Article information

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

Received: 22 July 2004
Accepted: 2 January 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 57R95: Realizing cycles by submanifolds

Twist spinning Seiberg–Witten invariants branched covers ribbon knots


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.

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