  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
$\mathcal{L}_{\gamma}(A)$.

Geometric Topology