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

TL;DR

This paper demonstrates that the hat and plus versions of the Heegaard Floer contact invariant can differ, revealing new phenomena in contact topology on Brieskorn spheres.

Contribution

It shows the existence of tight, zero-twisting contact structures with non-vanishing hat invariant but vanishing plus invariant, a previously unknown phenomenon.

Findings

Existence of tight, zero-twisting contact structures on Brieskorn spheres.

The hat invariant can be non-zero while the plus invariant is zero for the same contact structure.

Uncovered a new discrepancy between different versions of the Heegaard Floer contact invariant.

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.