Cutting and Gluing Surfaces

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.