= Polygon with a complete set of side-identifications
Start with a polygonal disk and pair all its sides, specifying for each pair a homeomorphism matching endpoints. Arrow labels encode whether each occurrence is traversed with or against its assigned direction. Form the <quotient topology> using those identifications and no extra identifications in the interior. Matched edge interior points have disk neighbourhoods formed from two half-disks; the corner sectors around each identified vertex form a circular link. The quotient is therefore a closed <surface>. A boundary word records each edge label exactly twice. Its endpoint identifications determine the vertex count, while its signs determine orientability. This encoding is distinct from treating a side-paired polygon as an abstract <simplicial complex>; a compatible subdivision is needed for a simplicial triangulation.
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