Heegaard Floer Invariants and Tight Contact Three--Manifolds

Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight contact structures on Y(r) for every r not equal to 1. This implies, in particular, that the oriented boundaries of the positive E6 and E7 plumbings carry positive, tight contact structures, solving a well-known open problem.