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 FigueroaO'Farrill Mar 8 '11 at 17:54
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8$\begingroup$ "Best" has no precise meaning when it comes to polynomial invariants of knots. $\endgroup$ – Ryan Budney Mar 8 '11 at 15:59

1$\begingroup$ By 'best' I meant: Is the HOMFLY polynomial better at discriminating knots then other polynomial invariant that are determine by skein relation. $\endgroup$ – dlb Mar 8 '11 at 16:45
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 2variable 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.

1$\begingroup$ Thanks for the references. I had not come across the coloured HOMFLYPT before. $\endgroup$ – dlb Mar 8 '11 at 17:35