# Differential relations for almost Belyi maps

## TL;DR

This paper explores differential relations for almost Belyi maps, utilizing Saito's free divisors and Kitaev's Painleve VI solutions to compute specific algebraic maps in genus 0 and 1.

## Contribution

It introduces new differential relations for almost Belyi maps and applies them to compute all genus 0 and 1 Painleve VI solutions in a specific classification.

## Key findings

- Derived differential relations for polynomial components of almost Belyi maps.
- Connected Saito's theory of free divisors to the structure of these maps.
- Computed all genus 0 and 1 Painleve VI solutions in the Lisovyy-Tykhyy classification.

## Abstract

Several kinds of differential relations for polynomial components of almost Belyi maps are presented. Saito's theory of free divisors give particularly interesting (yet conjectural) logarithmic action of vector fields. The differential relations implied by Kitaev's construction of algebraic Painleve VI solutions through pull-back transformations are used to compute almost Belyi maps for the pull-backs giving all genus 0 and 1 Painleve VI solutions in the Lisovyy-Tykhyy classification.