Infinitely many universally tight torsion free contact structures with vanishing Ozsváth–Szabó contact invariants

Abstract

Ozsváth–Szabó contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsváth–Szabó invariant vanishes. We also discuss the relation between these invariants and an invariant on T3 and construct other examples of new phenomena in Heegaard–Floer theory. Along the way, we prove two conjectures of K. Honda, W. Kazez and G. Matić about their contact topological quantum field theory. Almost all the proofs in this paper rely on their gluing theorem for sutured contact invariants.