Every compact connected surface comes from a standard polygon word, with unpaired edges recording boundary components.Suggest a slogan
Every connected closed surface is homeomorphic to the quotient of a polygon whose edges are identified in pairs according to one of the words \(a_1a_1^{-1}\), \(a_1b_1a_1^{-1}b_1^{-1}\cdots a_gb_ga_g^{-1}b_g^{-1}\) (\(g\geq 1\)), or \(a_1a_1\cdots a_ka_k\) (\(k\geq 1\)) [@Mun00]. For compact surfaces with boundary, the orientable surface of genus \(g\) with \(b\) boundary circles is obtained from the word \(\alpha_1\beta_1\alpha_1^{-1}\beta_1^{-1}\cdots\alpha_g\beta_g\alpha_g^{-1}\beta_g^{-1}\,\xi_1\rho_1\xi_1^{-1}\cdots\xi_b\rho_b\xi_b^{-1}\), in which the edges \(\rho_j\) are left unidentified and become the boundary circles.
