Kauffman-Jones polynomial of a curve on a surface

We introduce a Kauffman-Jones type polynomial ${\mathcal{L}}_{\gamma }(A)$ for a curve $\gamma$ on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial ${\mathcal{L}}_{\gamma }(A)$ is a Laurent polynomial in one variable $A$ and is an invariant of the homotopy class of $\gamma$. As an application, we obtain an estimate in terms of the span of ${\mathcal{L}}_{\gamma }(A)$ for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of ${\mathcal{L}}_{\gamma }(A)$ and show some computational results on the span of ${\mathcal{L}}_{\gamma }(A)$.