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Ozsvath-Szabo contact invariants are a powerful way to prove tightness ofcontact structures but they are known to vanish in the presence of Girouxtorsion. In this paper we construct, on infinitely many manifolds, infinitelymany isotopy classes of universally tight torsion free contact structures whoseOzsvath-Szabo invariant vanishes. We also discuss the relation between theseinvariants and an invariant on T^3 and construct other examples of newphenomena in Heegaard--Floer theory. Along the way, we prove two conjectures ofK Honda, W Kazez and G Matic about their contact topological quantum fieldtheory. Almost all the proofs in this paper rely on their gluing theorem forsutured contact invariants.

Infinitely many universally tight torsion free contact structures with vanishing Ozsv\'ath-Szab\'o contact invariants