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?



Set of Jones polynomials as the knot varies
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


* Seifert, H.; [Über das Geschlecht von Knoten](https://eudml.org/doc/159739). Math. Ann. 110 (1935), no. 1, 571–592.


