  We present some topological and arithmetical aspects of a class of Rauzy
fractals $\mathcal{R}_{a,b}$ related to the polynomials of the form
$P_{a,b}(x)=x^{3}-ax^{2}-bx-1$, where $a$ and $b$ are integers satisfying $-a+1
\leq b \leq -2$. This class has the property that $0$ lies on the boundary of
$\mathcal{R}_{a,b}$. We construct explicit finite automata that recognize the
boundaries of these fractals, which allows to establish the number of neighbors
of $\mathcal{R}_{a,b}$. In particular, we prove that if $2a+3b+4 \leq 0$ then
$\mathcal{R}_{a,b}$ is not homeomorphic to a topological disk. We also show
that the boundary of the set $\mathcal{R}_{3,-2}$ is generated by two infinite
iterated function systems.
