Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?

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    $\begingroup$ "...as the knot varies"? Varies how? If the variation is due to an ambient isotopy or the Reidemeister moves, then invariant polynomials are... well... invariant. $\endgroup$ – David G. Stork Jan 5 '19 at 18:39
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    $\begingroup$ As the knot ranges over the set of all knots; I believe this is clear from the content of my question, if not from the title. $\endgroup$ – pre-kidney Jan 5 '19 at 18:40

I believe your question is open for the Jones polynomial. However, it is solved for the Alexander polynomial. On page 171 of Rolfsen's Knots and Links, the following theorem appears.

Theorem. Let $p(t)$ be any Laurent polynomial satisfying:

  1. $p(1) = \pm 1$, and
  2. $p(t)=p(t^{-1})$.

There exists a knot $K$ whose Alexander polynomial $\Delta_K(t)$ is $p(t)$.

It is well-known that the Alexander polynomial satisfies the above conditions. In the proof, Rolfsen shows how to explicitly construct a knot $K$ whose Alexander polynomial is a given Laurent polynomial $p(t)$ satisfying the conditions above.

Rolfsen gives the original reference of this result as

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  • $\begingroup$ Thank you - even if the Jones polynomial question remains open, are you aware of any variant of Seifert's construction that produces many (if not all) Jones polynomials? $\endgroup$ – pre-kidney Jan 5 '19 at 20:35
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    $\begingroup$ I think that not much is really known about how to produce knots with prescribed Jones polynomial. Manchon arXiv:0201160 used a construction of Morton and Bae arXiv:0012089 to construct prime knots whose Jones polynomials have prescribed extreme coefficients. $\endgroup$ – Adam Lowrance Jan 5 '19 at 20:47

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