4-dimensional aspects of tight contact 3-manifolds

TL;DR

This paper proposes a 4-dimensional criterion for tight contact structures on 3-manifolds, linking it to a slice-Bennequin inequality and providing evidence through Heegaard Floer invariants, with implications for knot theory.

Contribution

It introduces a conjecture connecting tightness to a slice-Bennequin inequality and proves it for structures with non-zero Ozsváth-Szabó invariants, advancing understanding of contact topology.

Findings

Proved the conjecture for contact structures with non-vanishing Ozsváth-Szabó invariant.

Showed subsurfaces of open book pages maximize slice Euler characteristic under certain conditions.

Conjectured a broader relationship between tightness and open book decompositions.

Abstract

In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[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 Yx[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsv\'ath-Szab\'o contact invariant. We also show that any subsurface of a page of an open book inducing a contact structure with non-trivial invariant maximize "slice" Euler-characteristic for its boundary, and conjecture that this holds more generally for open books inducing tight contact structures.