Is the HOMFLY Polynomial the best polynomial invariant that can be calculation from skein relation?
closed as not constructive by Ryan Budney, Dan Petersen, Nate Eldredge, Ian Agol, José Figueroa-O'Farrill Mar 8 '11 at 17:54
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What about the coloured HOMFLYPT? It's clearly stronger than the HOMFLYPT. Whether it is a complete knot invariant is (I believe) open. Mutation preserves the HOMFLYPT polynomial. The 2-variable HOMFLYPT fares better, but also isn't a complete knot invariant. Examples of knots with the same coloured Jones polynomials (all colours), HOMFLYPT, and Kauffman polynomials, but possibly different coloured HOMFLYPT polynomials, are given in Proposition 1.5 HERE.
I agree that this is a badly posed question which should probably be closed. My motivation for answering is to advertise the references, which I think are lovely papers with closely related results.