| We study the colored Jones polynomials of links as defined using skein theory. The motivation and goal of this study is a conjecture relating these polynomials to the hyperbolic geometry of the complement of links.--- the volume conjecture.;This thesis consists of three parts. In the first part, we give a new proof of an expansion formula in the skein space. The proof starts with two simple geometric observations and then followed by a concrete calculation involving triple summations.;In the second part, using the formula obtained in the first part and following Kashaev's work on torus knots, we show that the volume conjecture is true for certain cable of the torus knots. That involves the study of the asymptotic behavior of a double integral. For function of two complex variables, change of path of integration is a different thing. One should also be careful about possible cancellation between terms coming from asymptotic expansion of different parts.;In the final part, we evaluate a trivalent graph in three sphere. Using this result we derive an expression of the colored Jones polynomials of the Borromean rings. This expression is useful to study the conjecture in this case. Our method works as well for a sequence of links obtained from the Borromean rings by taking branched covering of the three sphere. |
