Question

Any finite dimensional representation of $SL(n,\mathbb{C})$ can be found as a direct summand inside of some tensor product of the fundamental representations. Since we use finite dimensional representations of the quantum $SL(n)$ to construct the colored HOMFLYPT invariant one can naively hope that the invariants colored by the fundamental representations plus some independent of knot combinatorics would produce you colored HOMFLYPT for any color. How far from true is this naive expectation? Any suggestions or references are very welcomed.

Answer 1

It depends on what you call "independent of knot combinatorics". Obviously one can get the invariant associated with a representation $V_\lambda\subset V^{\otimes k}$, where $V$ is the fundamental representation, by replacing every strand of the link by $k$ strands (cabling) colored by $V$ and then inserting the projector onto $V_\lambda$; since this projector can be written in terms of Hecke algebra operators (this is the $q$-version of Young symmetrizer), it can be presented graphically as a certain linear combination of $k-k$ tangles. This is what is done in the paper cited by Scott. Not sure if this is the answer you are looking for.

Answer 2

The knot invariants given by the fundamental representations do not determine the knot invariant given by any representation. The fundamental representations have the property that the tensor square is multiplicity free and this implies that these invariants will not distinguish between mutant knots. However other representations will.

What I would expect is that if we fix a knot and consider the knots invariants indexed by partitions then this system is D-finite or holonomic. For a specific $n$ this was proved by Garoufalidis. I am not aware of any work on the two-variable system.