On a Finite Type Invariant Giving Complete Classification of Curves on Surfaces

In this paper, we construct a complete invariant for stably homeomorphic classes of curves on compact oriented surfaces without boundaries and show that this is a finite type invariant for curves. In knot theory, it is still unknown whether finite type invariants completely classify knots (the Vassiliev conjecture). We consider the analogy to this conjecture for generic immersed curves: do finite type invariants for immersed curves completely classify them? The answer is in this note and it is true for stably homeomorphic classes of curves on compact oriented surfaces without boundaries. That is to say, there is a finite type invariant for curves on surfaces gives complete classification of curves on surfaces. In this short note, the author exposes the main results and the idea of the proofs. The details in the systematic context from the background will be given elsewhere. 1. Curves. A curve is a smooth immersion from a oriented circle to a closed oriented surface. We equip the following definitions. First, a curve is generic if all of the singular points are transversal double points of self-intersection. Second, a curve is singular if all of the singular points are transversal double points, self-tangency and triple point of self-intersection. Third, a pointed curve is a curve endowed with marked base point except on the singular points. Two curves are stably homeomorphic if there is a homeomorphism of their regular neighborhoods in the ambient surfaces mapping the first curve onto the second one preserving the orientation of the curves and the surfaces. Similarly, two pointed curves are stably homeomorphic if two curves are stably homeomorphic preserving their marked base points. 2. The definition of finite type invariants for immersed curves. We call self-tangency or triple point singular point. For each singular point, the direction of the resolution of the singular point is called positive if the resolution generates the part with a larger number of double points or positive triangle where the sign of the triangle is the following. We denote by q the number of sides where this orientation coincides with the orientation of the curve. We define the sign of triangle by (−1) (Fig. 1). The direction of the resolution is called negative if the direction is non-positive. 1