MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is the HOMFLY Polynomial the best polynomial invariant that can be calculation from skein relation?

share|cite|improve this question

closed as not constructive by Ryan Budney, Dan Petersen, Nate Eldredge, Ian Agol, José Figueroa-O'Farrill Mar 8 '11 at 17:54

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

"Best" has no precise meaning when it comes to polynomial invariants of knots. – Ryan Budney Mar 8 '11 at 15:59
By 'best' I meant: Is the HOMFLY polynomial better at discriminating knots then other polynomial invariant that are determine by skein relation. – 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 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.

share|cite|improve this answer
Thanks for the references. I had not come across the coloured HOMFLYPT before. – dlb Mar 8 '11 at 17:35

Not the answer you're looking for? Browse other questions tagged or ask your own question.