In this note, we define a notion of finite-type for invariants of curves on surfaces as an analogue of the notion of finite-type for invariants of knots and 3-manifolds (Section 3). We also present a systematic construction for a large family of finite-type invariants SCIn for curves on surfaces (Section 5). Arnold's invariants of plane isotopy classes of plane curves occur as invariants of order 1. Our theory of finite-type invariants of curves on surfaces is developed using the topological theory of words.
"Finite-type invariants for curves on surfaces." Proc. Japan Acad. Ser. A Math. Sci. 85 (9) 129 - 134, November 2009. https://doi.org/10.3792/pjaa.85.129