Denote the free group on 2 letters by F_2 and the SL(2,C)-representation
variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by
conjugation. We construct an isomorphism between the coordinate ring C[SL(2,C)]
and the ring of matrix coefficients, providing an additive basis of
C[R]^SL(2,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 SL(2,C)-character
variety of F_2 and a new proof of a classical result of Fricke, Klein, and
Vogt.Comment: Updated historical treatment of the subject. Figures drawn with
PGF/TikZ; Handbook of Teichmuller Theory II, A. Papadopoulos (ed), EMS
Publishing House, Zurich, 200
