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.