Universal geometric coefficients for the once-punctured torus

Abstract

We construct universal geometric coefficients, over the integers, the rationals, and the reals, for cluster algebras arising from the once-punctured torus. We verify that the once-punctured torus has a property called the Null Tangle Property. The universal geometric coefficients over the integers and the rationals are then given by the shear coordinates of certain "allowable" curves in the torus. The universal geometric coefficients over the reals are given by the shear coordinates of allowable curves together with the normalized shear coordinates of certain other curves each of which is dense in the torus. We also construct the mutation fan for the once-punctured torus and recover a result of Nájera on g-vectors.

26 pages, 9 figures. Version 2: Minor expository changes. Version 3: Very minor expository changes. Version 4: Final version to appear in Séminaire Lotharingien de Combinatoire. Fixed a crucial typo in Proposition 3.1. Added two brief sections at the end, one discussing and picturing denominator vectors, and another discussing extensions to other surfaces and cluster algebras

References in corpus (4)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Universal geometric cluster algebras
• On the c-vectors and g-vectors of the Markov cluster algebra
• Tubular Cluster Algebras II: Exponential growth

Cited by in corpus (7)

• Universal geometric cluster algebras
• Initial-seed recursions and dualities for d-vectors
• Universal geometric cluster algebras from surfaces
• On Jacobian algebras associated with the once-punctured torus
• Universal geometric coefficients for the four-punctured sphere
• Cluster automorphisms and quasi-automorphisms
• Dominance phenomena: mutation, scattering and cluster algebras