Classical complex dynamics began with an interest in the topology and combinatorics of the moduli space of quadratic polynomials {z^2+c}, notable also for its special subset, the Mandelbrot set. The moduli space of all quadratic rational maps rat_2 is isomorphic to \mathbb{C}^2, and we can also understand the space of quadratic polynomials as a special curve within rat_2 in which one critical point is marked as a fixed point. Other natural curves within rat_2 of longstanding interest are the Per_n(0) curves, in which one critical point is marked as in an n-cycle. In this talk, we speak about the topology and combinatorics of these curves and their bifurcation locus, paying particular attention to how structure from the Mandelbrot set (“matings” and “captures”) can shed light on questions like Per_n(0) is irreducible.