Type: Article

Complex Polynomials and Tessellations in the Riemann Sphere

González-Cely, Leidy J.; Muciño-Raymundo, Jesús;

Abstract:

Let P be a complex polynomial of degree in the Riemann sphere, and let be an oriented Jordan path running through its critical values. A classical algorithm, rooted in the pioneering work of H. A. Schwarz and F. Klein, states that the inverse image of under P determines a finite tessellation of the Riemann sphere, with tiles that are topological k–polygons and alternate colors. Following a question by W. P. Thurston, we study under what conditions a finite graph (or equivalently a finite tessellation of the Riemann sphere) originates from a generic polynomial P and an oriented Jordan path as above.