Jones polynomial of the concatenation of two braids 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 $\sigma$. 
 A: In the very special case where $\sigma_2$ is a full twist of all $n$ strands, there is some information. This is essentially because full twists are in the center of the braid group. See this paper by Champanerkar and Kofman for details:
http://www.math.csi.cuny.edu/~ikofman/jp_gap_PAMS.pdf
A: 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.
