Let $\sigma\_1$ and $\sigma\_2$ be two braids with $n$-strings.


Are there any formulas relating $J\_{\widehat{\sigma\_1\sigma\_2}}(q)$, $J\_{\hat{\sigma\_1}}(q)$, and $J\_{\hat{\sigma\_2}}(q)$? 


Here, $J\_L(q)$ is the jones polynomial of a link $L$ and $\hat{\sigma}$ stands for the closure of a braid $\sigma$. 



Jones polynomial of the concatenation of two braids
Here is one reason not to expect such a relationship (although I'm not sure if it can be completed to a proof). The Jones polynomial $J\_\sigma$ (roughly) comes from taking the trace of a linear map $A\_\sigma$ associated to the braid $\sigma$, so the question (roughly) asks about relations between $Tr(A\_\sigma)$, $Tr(A\_\tau)$, and $Tr(A\_\sigma A\_\tau)$. 


Let $a,b$ generate a free group $\pi$, and let $Chr$ be the $SL\_2$ character scheme of $\pi$ (i.e. the space of pairs of $SL\_2$ matrices considered up to simultaneous conjugation). Define $Tr(g):Chr \to \mathbb C$ to be the function $\rho \mapsto Tr(\rho(g))$. Then the functions $Tr(a)$, $Tr(b)$, and $Tr(ab)$ are algebraically independent. So there shouldn't be any universal formula relating these three traces, whether $a,b \in SL\_2$ or are bigger matrices.


