Combinatorics of generic 5-degree polynomials

Abstract

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.