TITLE: Jones polynomial of the concatenation of two braids
QUESTION [5 upvotes]: 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 $\sigma$.

REPLY [6 votes]: 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