Abstract

Abstract. We construct universal geometric coefficients, over Z, Q, and R, 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 Z and Q are then given by the shear co-ordinates of certain “allowable ” curves in the torus. The universal geometric coefficients over R are given by the shear coordinates of allowable curves to-gether 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. Contents