  Recent work of Dylan Thurston gives a condition for when a post-critically
finite branched self-cover of the sphere is equivalent to a rational map. We
apply D. Thurston's positive criterion for rationality to give a new proof of a
theorem of Rees, Shishikura, and Tan about the mateability of quadratic
polynomials when one polynomial is in the main molecule. These methods may be a
step in understanding the mateability of higher degree post-critically finite
polynomials and demonstrate how to apply the positive criterion to classical
problems.
