The hat and plus version of the Heegaard Floer contact invariant are not equivalent

Abstract

We advance Matkovič ideas, originally applied to complete the classification of tight structures on small Seifert fibred $L$-spaces, to show the existence of contact structures on Brieskorn spheres which are tight and zero-twisting. This uncovers a phenomenon that has never appeared in literature before: namely, that a contact structure $\xi$ on a 3-manifold can be such that $\hat{c}(\xi )$ is non-vanishing, but $c^{+}(\xi )$ is zero.