# Jones polynomial of cable knots

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 or Seifert matrices?

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.