I am trying to understand the colored HOMFLY polynomials. The theoretic description Anna Aiston gave in her PhD thesis is really nice, but what about the computation?
I would like to understand the knots-quivers correspondence, so I would like to learn how one can compute the $S^r$-colored HOMFLY polynomials. How can the $S^r$-colored HOMFLY polynomials $\overline{P}_K$ of a given knot $K$ be computed? For example, in the easiest cases of the unknot or the trefoil, is there a certain way how to compute the formulas as stated in the paper about the knots-quivers correspondence?