Combinatorics of generic 5-degree polynomials

We consider the space P of generic complex 5-degree polynomials. Critical values of such polynomial, i.e. four points in the complex plane, either are vertices of a convex quadrangle Q, or vertices of a triangle T with one point inside T. The inverse image of Q is a tree-like connected structure of five ovals (a cactus). The inverse image of T is also a cactus, but of four ovals. Transformations of cacti of the first type into cacti of the second type and vice versa allow one to represent the space P as a ribbon bipartite graph of genus 3.