**Spin Networks and SL(2,C) Character Varieties** (with Sean Lawton), *Handbook of Teichmuller Theory Volume II* (Chapter 16), 2009, Athanase Papadopoulos, ed., EMS Publishing House. arXiv: math.QA/0511271

**Abstract:** Denote the free group on 2 letters by $\mathcal{F}_2$ and the $\mathrm{SL}(2,\mathbb{C})$-representation variety of $\mathcal{F}_2$ by $\mathcal{R}=\mathrm{Hom}(\mathcal{F}_2,\mathrm{SL}(2,\mathbb{C}))$. The group $\mathrm{SL}(2,\mathbb{C})$ acts on $\mathcal{R}$ by conjugation. We construct an isomorphism between the coordinate ring $\mathbb{C}[\mathcal{F}_2,\mathrm{SL}(2,\mathbb{C})]$ and the ring of matrix coefficients, providing an additive basis of $\mathbb{C}[\mathcal{R}]^{\mathrm{SL}(2,\mathbb{C})}$ in terms of spin networks. Using a graphical calculus, we determine the symmetries and multiplicative structure of this basis. This gives a canonical description of the regular functions on the $\mathrm{SL}(2,\mathbb{C})$-character variety of $\mathcal{F}_2$ and a new proof of a classical result of Fricke, Klein, and Vogt.