On the volume conjecture for polyhedra

We formulate a generalization of the volume conjecture for planar graphs. Denoting by the Kauffman bracket of the graph whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain evaluations of grow exponentially as and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is (up to a local modification around all the vertices) and with dihedral angles given by c. We provide evidence for it, by deriving a system of recursions for the Kauffman brackets of planar graphs, generalizing the Gordon-Schulten recursion for the quantum 6j-symbols. Assuming that does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schlafli equation); moreover, any small perturbation of the volume function that is still a solution to these equations, is a perturbation by an additive constant. In the appendix we also provide a proof outlined elsewhere of the conjecture for an infinite family of planar graphs including the tetrahedra.