Loop Polynomials For Knots on a 3-Sphere

We develop a method of calculating the first loop polynomials of a knot based on the $R$-matrix Reshetikhin-Turaev formula for the colored Jones and HOMFLY-PT polynomials. The method is based on (1) replacing the finite-dimensional $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ representations with Verma modules of the same highest weight, (2) identifying each Verma module with the polynomial algebra, one generator per positive root, (3) presenting each $R$-matrix as a composition of Burau-based linear transformation of polynomial variables and a series in $\ln q$ with coefficients being differential operators acting on the polynomials and, finally, (4) computing the traces of products of these operators by using a combinatorial trick commonly used in deriving the Feynman diagram rules. We present some computational results based on this method.