Planar multilinks and rational singularities

TL;DR

This paper characterizes rational surface singularities using planar multilinks and explores their topological properties, linking singularity theory with contact topology and Heegaard Floer invariants.

Contribution

It establishes a connection between planar multilinks supporting canonical contact structures and the rationality of surface singularities, extending topological properties to multilinks.

Findings

Rational singularities are characterized by planar multilinks supporting their contact structures.

Planar multilinks with a component of multiplicity 1 characterize sandwiched singularities.

Topological properties like negative definiteness of fillings extend from open books to multilinks.

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.