Heegaard floer invariants of contact structures on links of surface singularities

Let a contact 3-manifold.Y; Ɛ₀ / be the link of a normal surface singularity equipped with its canonical contact structure Ɛ₀. We prove a special property of such contact 3-manifolds of “algebraic” origin: the Heegaard Floer invariant c^C.Ɛ₀ / 2 HF^C. Y / cannot lie in the image of the U-action on HF^C. Y /. It follows that Karakurt’s “height of U-tower” invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of U-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and Némethi’s lattice cohomology.