Generalized volume and geometric structure of 3-manifolds

For several hyperbolic knots, a relation between certain quantum invariants and the volume of their complements are discovered by R. Kashaev in [2]. In [6], it is shown that Kashaev’s invariants are specializations of the colored Jones polynomials. Kashaev used the saddle point method to obtain certain limit of invariants, and Y. Yokota proved that the equations to determine the saddle points correspond to the equations defining the hyperbolic structure of the knot complement. He introduce a simplicial decomposition of the complement associated to a knot diagram, and show that the equations to give the hyperbolic structure of each simplex coincide with the equations for saddle points. For 3-manifolds obtained by surgeries along a figure-eight knot, H. Murakami [5] follows Kashaev’s computation for the Witten-Reshetikhin-Turaev invariants and found that a value at certain saddle point relates to the volume. Trying to extend these works to the Turaev-Viro invariant [9], a formula for the volume of a hyperbolic tetrahedron is obtained in [7]. The Turaev-Viro invariant is defined from a simplicial decomposition of a 3-manifold, and use a state sum associating the quantum 6j-symbol to each tetrahedron, and the formula for the volume of a hyperbolic tetrahedron comes from the quantum 6j-symbols. Moreover, extending Yokota’s theory to this case, we may get some relation between the volume and the geometric structure of the manifold, which is the main subject of this note.