An obstruction to sliceness via contact geometry and ?classical? gauge theory

Abstract

In recent work (beginning with [8]) Kronheimer and Mrowka have developed the formidable machinery of "gauge theory for embedded surfaces", and deployed it against a host of problems of the general form, "What is the least genus of a smooth surface representing a specified homology class in a given 4-manifold?" One of their results (the "local Thom Conjecture") can be used [19] to derive a lower bound (the "slice-Bennequin inequality", hereinafter sBi) for the Murasuoi (or slice) genus of a knot K c S3--that is, the smallest genus of a smooth, oriented surface in D 4 with boundary K. A knot is slice if it has Murasugi genus 0, so in particular sBi provides an obstruction to sliceness. Though Bennequin originally conjectured sBi in the context [1] of an investigation into 3-dimensional contact geometry, to date sBi has not been proved using exclusively 3-dimensional methods. (Such methods did success- fully establish [1] the "Bennequin inequality" pure and simple, which sBi generalizes. It appears they can also be used [I 1] to answer the "question of Milnor" on unknotting number, achieved 4-dimensionally - using the local Thom Conjecture - in [8].) The following corollary of sBi will be proved in w independently of sBi, by combining ideas from dimensions 3 (contact geometry) and 4 ("classical" gauge theory [4], earlier and comparatively simpler than [8]): ifa knot K has non-negative maximal Thurston-Bennequin invariant TB(K), then K is not slice. Though TB (K) is defined analytically, an alternative combinatorial definition (recalled in w allows one to show TB(K) > 0 (and sometimes to calculate TB(K)) for many interesting knots K; in particular, results of [19] on non- sliceness of certain iterated doubled knots are recovered quickly and easily. Research partially supported by CNRS.