Gluings of Surfaces with Polygonal Boundaries

By pairwise gluing edges of a polygon, one obtains two-dimensional surfaces with handles and holes. We compute the number $\mathcal{N}_{g,L}(n_1,\dots,n_L)$ of distinct ways to obtain a surface of given genus $g$ whose boundary consists of $L$ polygonal components with given numbers $n_1,\dots,n_L$ of edges. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursion relations between the $\mathcal{N}_{g,L}$. We show that the Harer–Zagier numbers arise as a special case of $\mathcal{N}_{g,L}$ and derive a new closed-form expression for them.