Reidemeister torsion, twisted Alexander polynomial, the A-polynomial, and the colored Jones polynomial of some classes of knots

This dissertation studies invariants of knots and links.; In Chapter 1 we study a twisted Alexander polynomial of links in the projective space R P3 using its identification with Reidemeister torsion. We prove a skein relation for this polynomial.; Chapter 2 studies relationships between the A-polynomial of a 2-bridge knot and a twisted Alexander polynomial associated with the adjoint representation of the fundamental group of the knot complement. We show that for twist knots the A-polynomial is a factor of the twisted Alexander polynomial.; Chapter 3 studies the irreducibility of the A-polynomial of 2-bridge knots. We show that the A-polynomial A(L, M) of a 2-bridge knot b (p, q) is irreducible if p is prime, and if (p - 1)/2 is also prime and q ≠ 1 then the L-degree of A(L, M) is (p - 1)/2. This shows that the AJ conjecture relating the A-polynomial and the colored Jones polynomial holds true for these knots, according to work of Le.; In Chapter 4 a determinant formula for the colored Jones polynomial is obtained. This determinant formula is similar to the known determinant formula for the volume of a hyperbolic knot obtained via L 2-torsion. This study is in the context of the volume conjecture relating the colored Jones polynomial to the hyperbolic volume of a knot.; Major parts of this dissertation are joint works with Thang T. Q. Le.