Affiliation: College of Arts and Sciences, Department of Mathematics
Both the Alexander polynomial and the colored Jones polynomial are well-known knot invariants. While the Jones polynomial seems similar to the Alexander polynomial, it lacks an interpretation in classical topology. Because the Alexander polynomial has a classical topological definition, exploring a relationship between the two polynomials offers the possibility of interpreting the Jones polynomial topologically. Melvin and Morton conjectured a relationship between the two through an expansion of the colored Jones polynomial. The conjecture was proven by Bar-Natan and Garoufalidis and Rozansky extended the result further. Rozansky proved an expansion of the colored Jones polynomial in h=q-1. At each power of h in the expansion, there is a rational expression with powers of the Alexander polynomial in the denominator and new polynomial knot invariants in the numerator. In this dissertation, we will describe how we used the quantum group of sl?2? and techniques from quantum field theory to calculate the first two of these polynomial invariants for all prime knots of up to nine crossings and present these results. Furthermore, we will provide evidence of the validity of a conjecture from Rozansky by calculating the first two polynomial invariants in the expansion for all amphicheiral knots of up to ten crossings.