Counting holomorphic curves in a symplectic manifold has
been an area of research since Gromov’s work on this subject in the
1980s. Symplectic manifolds naturally allow a ‘cut’ operation. We
explore what happens to curve count-based invariants when a collection
of cuts is applied to a symplectic manifold. An interesting feature
of curves in a multiply-cut manifold is that they have an underlying
‘tropical graph’, which is a graph that lives in the polytope
associated to the cut.