This dissertation studies the colored Jones polynomial of knots and links, colored by representations of simple Lie algebras, and the stability of its coefficients. Chapter 1 provides an explicit formula for the second plethysm of an arbitrary representation of sl3. This allows for an explicit formula for the colored Jones polynomial of the trefoil, and more generally, for T(2,n) torus knots. This formula for the sl3 colored Jones polynomial of T(2,n)$ torus knots makes it possible to verify the Degree Conjecture for those knots, to efficiently compute the sl3 Witten-Reshetikhin-Turaev invariants of the Poincare sphere, and to guess a Groebner basis for the recursion ideal of the sl3 colored Jones polynomial of the trefoil. Chapter 2 gives a formulation of a stability conjecture for the coefficients of the colored Jones polynomial of a knot, colored by irreducible representations in a fixed ray of a simple Lie algebra. The conjecture is verified for all torus knots and all simple Lie algebras of rank 2. Chapter 3 supplies an efficient method to compute those q-series that come from planar graphs (i.e., reduced Tait graphs of alternating links) and compute several terms of those series for all graphs with at most 8 edges. In addition, a graph-theory proof of a theorem of Dasbach-Lin which identifies the coefficient of q^k in those series for k=0,1,2 in terms of polynomials on the number of vertices, edges and triangles of the graph is given. Chapter 4 provides a study of the structure of the stable coefficients of the Jones polynomial of an alternating link.The first four stable coefficients are identified with polynomial invariants of a (reduced) Tait graph of the link projection. A free polynomial algebra of invariants of graphs whose elements give invariants of alternating links is introduced which strictly refines the first four stable coefficients. It is conjectured that all stable coefficients are elements of this algebra, and experimental evidence for the fifth and sixth stable coefficient is given. The results are illustrated in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.

This thesis has four chapters. After a brief introduction in Chapter 1, the $AJ$-conjecture is introduced in Chapter 2. The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If $K$ satisfies the $AJ$-conjecture, sufficient conditions on $K$ are given for the $(r,2)$-cable knot $C$ to also satisfy the $AJ$-conjecture. If a reduced alternating diagram of $K$ has $\eta_+$ positive crossings and $\eta_-$ negative crossings, then $C$ will satisfy the $AJ$-conjecture when $(r+4\eta_-)(r-4\eta_+)>0$ and the conditions of Theorem 2.2.1 are satisfied. Chapter 3 is about quantum curves and their relation to the $AJ$ conjecture. The variables $l$ and $m$ of the $A$-polynomial are quantized to operators that act on holomorphic functions. Motivated by a heuristic definition of the Jones polynomial from quantum physics, an annihilator of the Chern-Simons section of the Chern-Simons line bundle is found. For torus knots, it is shown that the annihilator matches with that of the colored Jones polynomial. In Chapter 4, a tangle functor is defined using semicyclic representations of the quantum group $U_q(sl_2)$. The semicyclic representations are deformations of the standard representation used to define Kashaev's invariant for a knot $K$ in $S^3$. It is shown that at certain roots of unity the semicyclic tangle functor recovers Kashaev's invariant.

This dissertation studies quantum invariants of knots and links, particularly the colored Jones polynomials, and their relationships with classical invariants like the hyperbolic volume and the A-polynomial. We consider the volume conjecture that relates the Kashaev invariant, a specialization of the colored Jones polynomial at a specific root of unity, and the hyperbolic volume of a link; and the AJ conjecture that relates the colored Jones polynomial and the A-polynomial of a knot. We establish the AJ conjecture for some big classes of two-bridge knots and pretzel knots, and confirm the volume conjecture for some cables of knots.