Reidemeister torsion, twisted Alexander polynomial, the A-polynomial, and the colored Jones polynomial of some classes of knots
Huynh, Vu Quang
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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 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 [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 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.
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