Symmetric cubic polynomials

We describe a model \(\mathcal{M}_3^{comb}\) for the boundary of the connectedness locus \(\mathcal{M}^{sy}_3\) of the parameter space of cubic symmetric polynomials \(p_c(z)=z^3-3c^2z\). We show that there exists a monotone continuous function \(\pi:\partial \mathcal{M}_c^{sy}\to \mathcal{M}_3^{comb}\) which is a homeomorphism if \(\mathcal{M}^{sy}_3\) is locally connected.