4-dimensional aspects of tight contact 3-manifolds

Abstract

In this article we conjecture a 4-dimensional characterization of tightness: a contact structure on a 3-manifold Y is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Y × [0, 1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: if a fibered link L induces a tight contact structure on Y then its fiber surface maximize Euler characteristic amongst all surfaces in Y ×[0, 1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsváth-Szabó contact invariant.