Parametrization for a class of Rauzy Fractal

In this paper, we study a class of Rauzy fractals ${\mathcal{R}}_a$ given by the polynomial $x^3-a{x}^2+x-1$ where $a\ge 2$ is an integer. In particular we give explicitly an automaton that generates the boundary of ${\mathcal{R}}_a$ and using an exotic numeration system we prove that ${\mathcal{R}}_a$ is homeomorphic to a topological disk.