OK, the heading was a bit tersely formulated...
If you have a quantum group and an irrep, you theoretically know the R matrix (mathematicians are a notoriously idle lot, they give the general formula and thus the problem is solved :-) - and the characteristic equation of the R matrix is a valid skein equation.
Now to the reverse process. Question 1: Are there really skein
equations that can't be modeled with a R matrix? E.g. I heard that
already the Kauffman 2-variable polynomial is unattainable this way,
but I only heard it and never saw an actual proof. (I have no idea how the
situation is for directed knots.)
Question 2. OK, assume we have a working R matrix, is there always a quantum group associated with that? ("Baxterization"??) Again, I think the answer is "no" for unoriented knots. (If no quantum group-based knot polynomial can distinguish mutants - also from the "so I heard" variety - the question is solved, since I have a R matrix doing this.)