On conjectures related to character varieties of knots and Jones polynomials

It is well known that the Kauffman Bracket Skein Module of a knot complement K_q(S^3 \ K) is canonically a module over the Z_2-invariants of the quantum torus, A_q^{Z_2}, and this module determines the colored Jones polynomials J_n(K; q) of the knot K. Berest and Samuelson identified a conjecture for knots under which a close variant of K_q(S^3 \ K) canonically becomes a module over a certain Double Affine Hecke Algebra, from which they defined a family of polynomials J_n(K; q; t_1; t_2) generalizing the classical polynomials of Jones. In this thesis an analogue of Habiro’s cyclotomic equation for the J_n(K; q) is discovered for J_n(K; q; t_1; t_2). An integrality result for the coefficients in this equation is found as a corollary, offering evidence for the conjecture of Berest and Samuelson for all knots. Separately, the conjecture of Berest and Samuelson is studied at the particular value q = -1 where it is known to relate to properties of SL_2(C)-character varieties of knots. Computational methods are used to establish that the conjecture holds for some non-invertible knots, which was not previously known