Drawing Bers Embeddings of the Teichmüller Space of Once-Punctured Tori

We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmüller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. To each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide if the representation is discrete or not by applying the Jørgensen’s theory on the quasifuchsian space of once-punctured tori.