An obstruction to sliceness via contact geometry and “classical” gauge theory. (English) Zbl 0843.57011

The author shows that if a knot has a non-negative maximal Thurston-Bennequin invariant, then it is not slice. The method is a combination of 3-dimensional contact geometry and 4-dimensional “classical” gauge theory à la Donaldson. This is supposed to be in contrast to the new gauge theoretic techniques of Kronheimer and Mrowka which lead to the proof of the Thom conjecture and the Bennequin inequality for the slice genus of a knot. This inequality implies the first mentioned result. As a nice corollary of his result the author mentions that the simplest Casson-handle (with one kink in every step) is non-standard if all kinks have the same sign. This result was first obtained by Ž. Bižaca and is conjectured for arbitrary signs of the kinks.
Reviewer: P.Teichner (Mainz)


57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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