On the volume conjecture for polyhedra

Abstract

We formulate a generalization of the volume conjecture for planar graphs. Denoting by $$\langle \Gamma , c \rangle ^{\mathrm {U}}$$ the Kauffman bracket of the graph $$\Gamma $$ whose edges are decorated by real “colors” c, the conjecture states that, under suitable conditions, certain evaluations of $$\langle \Gamma ,\lfloor kc \rfloor \rangle ^{\mathrm {U}}$$ grow exponentially as $$k\rightarrow \infty $$ and the growth rate is the volume of a truncated hyperbolic hyperideal polyhedron whose one-skeleton is $$\Gamma $$ (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 $$\langle \Gamma ,\lfloor kc \rfloor \rangle ^{\mathrm {U}}$$ does grow exponentially these recursions provide differential equations for the growth rate, which are indeed satisfied by the volume (the Schläfli 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.