Geometric intersection numbers on a four-punctured sphere

Let ${\cal G}_{4}$ be the space of all simple closed geodesics on the punctured sphere $\Sigma _{4}$. We construct an explicit homeomorphism of the completion of ${\cal G}_{4}$ onto a circle by using geometric intersection numbers. Also, we relate these geometric intersection numbers to trace polynomials of transformations corresponding to geodesics in ${\cal G}_{4}$ in a representation of $\pi _{1}(\Sigma _{4})$ into $PSL(2,\bold{C})$.