Tessellations of rational complex functions and the Riemann's existence theorem

TL;DR

This paper explores how rational functions induce tessellations on Riemann surfaces and demonstrates a correspondence between such tessellations and the functions, providing a visual and topological characterization.

Contribution

It establishes a bi-directional relationship between rational functions and tessellations on Riemann surfaces, extending the understanding of their geometric and topological properties.

Findings

Rational functions of degree n induce specific tessellations with 2n tiles.

Any tessellation with certain properties corresponds to a rational function on some Riemann surface.

The tessellation provides a visual and topological description of the rational function.

Abstract

A complex rational function R, of degree n>1, on a compact Riemann surface M provided with a cyclic order of its q critical values, determines an homogeneous tessellation of the Riemann surface M, whose 2n tiles are topological q-gons with alternating colors.The tessellation provides a simple and straighforward visual description of the rational function R. Conversely, assume a possibly non homogeneous tessellation T of a compact differentiable surface M' with tiles of alternating colors and a suitable labelling in the vertices of its tiles. Non homogeneous means that the tiles of T are r-gons, for different values of r. Then there exists a Riemann surface structure M on M', a complex rational function R and a cyclic order of its critical values, such that the tessellation of R on M topologically coincides with the original T.