Let $K\_{p,q}$ be a $(p,q)$-cable of the non-trivial knot $K$ in $S^3$.


Is there a closed formula for the Jones polynomial for $K\_{p,q}$ as in the case of [Alexander polynomial](https://mathoverflow.net/questions/123141/vassilliev-invariants-of-knots-and-their-cables) or [Seifert matrices](https://mathoverflow.net/questions/368955/seifert-matrices-of-cable-knots)?



Jones polynomial of cable knots
As Ian Agol mentioned, if there were a closed formula for the Jones polynomial $V(K\_{p,q})$ in terms of $V(K)$, this would give a closed formula for the colored Jones polynomials $V\_n(K)$ in terms of the original Jones polynomial $V(K) = V\_2(K)$.


However, this makes me think that there is no such simple formula. If there were, then we'd be able to easily give closed formulas for $V\_n(K)$ for arbitrary $n$, but these are usually quite hard to generate. It's one reason why the Volume Conjecture is only known to hold in special cases: the first step of a proof for $K$ [1] is usually to give a closed formula for $V\_n(K)$.


[1] Recently there have been proofs for other knots related to "fundamental shadow links" in $\#^k S^2 \times S^1$, which proceed differently. This isn't terribly relevant to your question, but I'm mentioning it for completeness.


