Recent Volumes 5: Gauge Theory and Low-Dimensional Topology 4: ANTS XIV 3: Hillman: Poincaré Duality 2: ANTS XIII 1: ANTS X
 The Open Book Series All Volumes About the Series Ethics Statement Purchase Printed Copies Author Index ISSN (electronic): 2329-907X ISSN (print): 2329-9061 MSP Books and Monographs Other MSP Publications
On uniqueness of symplectic fillings of links of some surface singularities

### Olga Plamenevskaya

Vol. 5 (2022), No. 1, 269–284
##### Abstract

We consider the canonical contact structures on links of rational surface singularities with reduced fundamental cycle. These singularities can be characterized by their resolution graphs: the graph is a tree, and the weight of each vertex is no greater than its negative valency. The contact links are given by the boundaries of the corresponding plumbings. In a joint work with L. Starkston, we have previously shown 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 and van Straten’s description of smoothings of these singularities. Here, we give a short self-contained proof of the uniqueness of fillings, via analysis of positive monodromy factorizations for planar open books supporting these contact structures.