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.)



Knot polynomials: Skein>Matrix>Group?
The Kauffman 2-variable knot polynomial probably can't be obtained from a quantum group if by this you mean the usual q-deformed universal enveloping algebras. If your two variables are $(r,q)$ and $r=\pm q^n$ then it can be obtained from quantum groups of type B,C or D (depending on $n$), see Wenzl's paper Comm. Math. Phys. 133 (1990) 383-432.


Generally a nice paper in which this kind of knot-invariant-to-quantum-group is discussed in Turaev and Wenzl's paper Math. Annalen 309 (1997), 411-461.


If you just start with an R-matrix and you want to define knot-invariants you need what Turaev calls an "enhanced Yang-Baxter operator." Given an R-matrix (enhanced or otherwise) it is possible to construct a braided Hopf algebra via the FRT (Faddeev-Reshetikhin-Takhtajan) construction but this will be unpleasant in general (infinite dimensional for sure).


