= Polygonal presentation of a punctured surface
Start with a <polygonal schema> whose outer boundary has $m$ sides, and remove a triangular disc strictly inside its polygon. Cut the resulting <annulus> along an arc from an outer vertex to an inner vertex. The cut space is a <closed disc> with $m+5$ marked sides: the original $m$ outer sides, two banks of the cut, and three inner sides. Parametrize it by a <convex polygon> with that many sides. A boundary <homeomorphism> carrying the marked vertices in cyclic order extends over the disc by radial extension after identifying both discs with a standard <closed disc>.
Transport the original side identifications to this polygon and identify the two cut banks, leaving the three inner sides unpaired. The <quotient space> is the original <surface> with an open triangular disc removed. In boundary-word notation, if $w$ is the original word, the new word is $w\,c\,d\,e\,f\,c^{-1}$, with $d,e,f$ unpaired. The <torus> has $w=aba^{-1}b^{-1}$; a square model of the <real projective plane> has $w=abab$, pairing antipodal points of the square's boundary.
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