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

Abstract

We advance Matkovi\v{c} 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 $\widehat c(\xi)$ is non-vanishing, but $c^+(\xi)$ is zero.