On a Closed Binding Curve of One-holed Torus

Given a closed binding curve $\gamma$ of a surface $\mathrm{\Sigma }$, any equivalence class of marked complete hyperbolic structure can be decomposed into polygons(possibly with a puncture) with sides being hyperbolic geodesic segments. When $\mathrm{\Sigma }$ is a one-holed torus and $\gamma =A^3{B}^2$, we show that any equivalence class of marked complete hyperbolic structure gives rise to an equilateral bigon with a puncture and a hexagon with equal opposite sides. In particular, we give a new coordinates of the Fricke Space of the one-holed torus.