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?
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:
- $p(1) = \pm 1$, and
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. Math. Ann. 110 (1935), no. 1, 571–592.