Planar multilinks and rational singularities

Abstract

Fibered multilinks are a generalization of classical fibered knots and open books that arise in the study of surface singularities and Milnor fibrations. We prove that if the canonical contact structure on the link of a surface singularity is supported by a planar multilink open book, then the singularity must be rational, and that sandwiched singularities are characterized by admitting planar multilinks with a component of multiplicity 1. We also show that some topological properties of planar open books extend to planar multilinks: symplectic fillings are negative definite and cannot contain symplectic surfaces of positive genus, and the image of the Heegaard Floer contact invariant vanishes in $HF_{red}$. Our results for singularities are based on these topological considerations, partly using Min--Roy--Wang's work on fillings of planar spinal open books, as well as the combinatorics of lattice embeddings.