  We study Weinstein 4-manifolds which admit Lagrangian skeleta given by
attaching disks to a surface along a collection of simple closed curves. In
terms of the curves describing one such skeleton, we describe surgeries that
preserve the ambient Weinstein manifold, but change the skeleton. The surgeries
can be iterated to produce more such skeleta --- in many cases, infinitely many
more.
  Each skeleton is built around a Lagrangian surface. Passing to the Fukaya
category, the skeletal surgeries induce cluster transformations on the spaces
of rank one local systems on these surfaces, and noncommutative analogues of
cluster transformations on the spaces of higher rank local systems. In
particular, the problem of producing and distinguishing such Lagrangians maps
to a combination of combinatorial-geometric questions about curve
configurations on surfaces and algebraic questions about exchange graphs of
cluster algebras.
  Conversely, this expands the dictionary relating the cluster theory of
character varieties, positroid strata, and related spaces to the symplectic
geometry of Lagrangian fillings of Legendrian knots, by incorporating cluster
charts more general than those associated to bicolored surface graphs.
