2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/171411This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-order invariants for plane immersed curves. The invariants in this theory are developed using the word theory proposed by Turaev.8 pages, The earlier versions contain mistakes at the title and Theorem 5.2 in ver 2 (: Theorem 4.2 in ver 1)Geometric Topology57M99text