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
, 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.
