Universal geometric coefficients for the four-punctured sphere

TL;DR

This paper constructs universal geometric coefficients for the cluster algebra of the four-punctured sphere, providing explicit shear coordinates and proving the Null Tangle Property for this surface.

Contribution

It introduces a classification of allowable curves, constructs universal coefficients, and proves the Null Tangle Property for the four-punctured sphere.

Findings

Constructed universal geometric coefficients for the four-punctured sphere.

Proved the Null Tangle Property for this surface.

Explicitly computed shear coordinates of allowable curves.

Abstract

We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the g-vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute shear coordinates explicitly to obtain universal geometric coefficients.