We introduce coordinate systems to the Teichmüller space of the twice-punctured torus and give matrix representations for the points of Teichmüller space. The coordinate systems allow representation of the mapping class group of the twice punctured torus as a group of rational transformations and provide several applications to the mapping class group and also to Kleinian groups.
"Teichmüller space and the mapping class group of the twice punctured torus." J. Math. Soc. Japan 73 (4) 1221 - 1252, October, 2021. https://doi.org/10.2969/jmsj/84998499