The complement of a fully augmented hyperbolic link is known to decompose into two identical, totally geodesic, right-angled ideal polyhedra, denoted by P. Let K be a hyperbolic link obtained by performing (p, 1) surgery on the crossing circles. Then, the complement of K can be viewed as the union of two identical deformations of P, which we call the "complexified polyhedron." Here, the edges are parameterized by the eigenvalues of certain elements in the knot group. This approach suggests a new method for proving the volume conjecture by using the Kirillov–Reshetikhin invariant of spatial graphs.