Heegaard Floer invariants of contact structures on links of surface singularities

Abstract

Let a contact 3-manifold (Y, \xi_0) be the link of a normal surface singularity equipped with its canonical contact structure \xi_0 . We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant c^+(\xi_0)\in \mathrm {HF}^+(-Y) cannot lie in the image of the U -action on \mathrm {HF}^+(-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.