Another example of a 3-manifold without tight contact structures

Abstract

It is known that every 3-manifold admits a contact structure. Since overtwisted contact structures were completely classified by Y. Elishberg, we are left with the classification of tight contact structures on 3-manifolds. Until now, only the Brieskorn homology spheres ∑(2,3,5) and -∑(2,3,3) are known not to admit any tight contact structures. In this paper we show that there is no tight contact structure on -∑(2,3,4) using the methods developed by Ko Honda and John Etnyre.