On a Closed Binding Curve of One-holed Torus

Given a closed binding curve $γ$ of a surface $Σ$, any equivalence class of marked complete hyperbolic structure can be decomposed into polygons(possibly with a puncture) with sides being hyperbolic geodesic segments. When $Σ$ is a one-holed torus and $γ= A^3 B^2$, 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.

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