3-valent graphs and the Kauffman bracket (Q1329003)

Given a 3-valent graph in the 3-sphere with edges colored with irreducible representations of SU(2) so that three edges meeting at a vertex form an admissible triple, one can define an invariant by using the Kauffman bracket. If the graph is a link this invariant coincides with the colored Jones polynomial as a framed link invariant. In this paper, the authors calculate the trihedron coefficient (the value of the planar theta-curve graph), the tetrahedron coefficient (the value of the planar tetrahedron graph), and the half-twist coefficient (the value multiplied after half-twisting at a vertex) by a graphical way. These coefficients are used to define the quantum \(6j\)-symbol.