Cluster topography

TL;DR

This paper reinterprets Conway's topograph using cluster algebra concepts, enabling new insights into analytic continuation, quadratic forms, and rational bijections through mutation rules and the Laurent phenomenon.

Contribution

It introduces a cluster algebra framework for topography, extending Conway's original ideas and applying them to Painlevé VI and quadratic form reduction.

Findings

Cluster construction of topograph via LP algebra toolkit

Analytic continuation for Painlevé VI exhibits Laurent phenomenon

Bijection between snake graphs and all rationals established

Abstract

Using the LP algebraic toolkit, Conway's original topograph is rethought of as a cluster construction, paving the way for a wider topography based on mutation-type local rules. As a remarkable application of such cluster-driven upgrade, both the process of analytic continuation for Painlev\'e VI and the reduction algorithm for quadratic forms are endowed with the Laurent phenomenon. En passant, the rattlesnake is defined so to complete the bijection between snake graphs and rationals to the whole of $\mathbb{Q}$.