Matings of cubic polynomials with a fixed critical point. Part II: <i>α</i>-symmetry of limbs

In this article we provide a combinatorial sufficient (and conjecturally, necessary) condition (called $\alpha$-symmetry) for the mating of two postcritically finite polynomials in $\mathcal{S}_1$ to be obstructed. To do this, study rotation sets associated parameter limbs connectedness locus $\mathcal{S}_1$, which allows us determine when there exist ray classes formal contain closed loop. We give proof necessity $\alpha$-symmetry particular subset maps $\mathcal{S}_1$. Many examples are given illustrate results paper.