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Séminaire Lotharingien de Combinatoire, B71e (2015), 30 pp.

# Nathan Reading

# Universal Geometric Coefficients for the Once-Punctured Torus

**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 coordinates of certain "allowable"
curves in the torus.
The universal geometric coefficients over **R** 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.

Received: March 4, 2014.
Accepted: January 23, 2015.
Final Version: January 29, 2015.

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