  We start with a disk with $2n$ vertices along its boundary where pairs of
vertices are connected with $n$ strips with certain restrictions. This forms a
{\it pairing}. To relate two pairings, we define an operator called a
cut-and-glue operation. We show that this operation does not change an
invariant of pairings known as the {\it signature.} Pairings with a signature
of $0$ are special because they are closely related to a topological
construction through cut and glue operations that have other applications in
topology. We prove that all balanced pairings for a fixed $n$ are connected on
a surface with any number of boundary components. As a topological application,
combined with works of Li, this shows that a properly embedded surface induces
a well-defined grading on the sutured monopole Floer homology defined by
Kronheimer and Mrowka.
