= Topology of disk tiles glued along disjoint boundary arcs
{title2=$\chi=n-e,\quad\pi_1\cong F_{e-n+1}$}
Consider a connected assembly of $n$ <closed discs> formed by identifying $e$ pairs of boundary arcs, each arc used once and all arcs in a given tile having disjoint endpoints. Its gluing <graph> has one vertex per tile and one edge per paired arc. A <spanning tree> joins the initial discs into a single <closed disc>; each of the remaining $e-n+1$ pairings attaches a band. This is a thickening of the gluing <graph>, so it is <homotopy equivalent> to that <graph>. Thus its <Euler characteristic> is $n-e$ and its <fundamental group> is the <free group> $F_{e-n+1}$. If the gluing <graph> is a tree, the assembled surface is itself a <closed disc>.
Suppose each pairing preserves the positive boundary parameter of the reference discs. A global <orientation> must then assign opposite orientation signs to adjacent tiles, because their induced <boundary orientations> on an interior seam must oppose one another. Such signs exist exactly when the gluing <graph> is <bipartite>. An odd gluing cycle therefore proves that the assembly is a <nonorientable surface>. A <Dirichlet boundary condition> or a <Neumann boundary condition> on an unpaired arc has no effect on this topological calculation.
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