Reidemeister torsion, twisted Alexander polynomial, the Apolynomial, and the colored Jones polynomial of some classes of knots
Abstract
This dissertation studies invariants of knots and links. In Chapter 1 we study a twisted Alexander polynomial of links in the projective space [Special characters omitted.] <math> <f> <blkbd>R</blkbd></f> </math> P 3 using its identification with Reidemeister torsion. We prove a skein relation for this polynomial. Chapter 2 studies relationships between the A polynomial of a 2bridge 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 2bridge knots. We show that the A polynomial A ( L, M ) of a 2bridge knot [Special characters omitted.] <math> <f> <ge>b</ge></f> </math> ( p, q ) is irreducible if p is prime, and if ( p  1)/2 is also prime and q ≠ 1 then the Ldegree 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.
Related items
Showing items related by title, author, creator and subject.

Fully Polynomial Time Approximation Scheme in Scheduling Deteriorating Jobs
U, Yixin; Cai, Pu; Cai, JinYi (1994) 
Frobenius's Degree Formula and Toda's Polynomials
Cai, JinYi (1995) 
Averagecase Complexity Theory and Polynomialtime Reductions
Pavan, A. (2001)