Kauffman-Jones polynomial of a curve on a surface

Abstract

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