On uniqueness of symplectic fillings of links of some surface singularities

TL;DR

This paper proves the uniqueness of symplectic fillings for certain links of surface singularities with specific resolution graph properties, using monodromy factorizations of supporting open books.

Contribution

It provides a self-contained proof of symplectic filling uniqueness for links of rational surface singularities with particular graph conditions, extending previous results.

Findings

Uniqueness of symplectic fillings for singularities with vertex weight ≤ -5

Use of positive monodromy factorizations in the proof

Simplified, self-contained proof approach

Abstract

We consider the canonical contact structures on links of rational surface singularities with reduced fundamental cycle. These singularities can be characterized by their dual resolution graphs: the graph is a tree, and the weight of each vertex is no greater than its negative valency. In a joint work with Starkston, we previously showed that if the weight of each vertex in the graph is at most -5, the contact structure has a unique symplectic filling (up to symplectic deformation and blow-up). The proof was based on a symplectic analog of de Jong-van Straten's description of smoothings of these singularities. In this paper, we give a short self-contained proof of uniqueness of fillings, via analysis of positive monodromy factorizations for planar open books supporting these contact structures.