UC San Diego

The colored Jones polynomial assigns to each knot a sequence of Laurent polynomials. This dissertation will focus on the patterns in the coefficients of these polynomials. We will discuss a new formula for calculating the colored Jones polynomial of certain pretzel knots and the stabilization and higher-order stabilization of the coefficients, specifically discussing what the second N coefficients of the N colored Jones polynomial of certain knots stabilize to. We will also look at patterns in the middle coefficients and explore a new way of looking at the colored Jones polynomials of amphichiral knots