Heegaard Floer invariants of contact structures on links of surface singularities

TL;DR

This paper proves that the Heegaard Floer invariant of canonical contact structures on links of surface singularities has a unique property, not lying in the U-action image, contrasting with general fillable structures.

Contribution

It establishes a special property of Heegaard Floer invariants for canonical contact structures on surface singularity links, linking Floer homology with lattice cohomology.

Findings

Heegaard Floer invariant $c^+(\xi_0)$ does not lie in the U-action image.

Karakurt's U-tower height invariants are always zero for these structures.

Contrasts with arbitrary U-tower heights in general fillable contact structures.

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 $c^+(\xi_0)\in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. 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 N\'emethi's lattice cohomology.