Let $V(L)$ be the Jones polynomial of the oriented link $L$. For $\alpha \in B_n$, we write $V(\alpha)$ for $V(\hat{\alpha})$, where $\hat{\alpha}$ is the closed braid associated to $\alpha$. The following questions should have immediate answers, but I'm nervous because I've never seen them written down.
1) Is $V(\alpha \beta \alpha^{-1})=V(\beta)$? Should be, since the two closed braids $\widehat{\alpha \beta \alpha^{-1}}$ and $\hat{\beta}$ are equivalent by a Markov move. 2) Is $V(\alpha\beta)=V(\beta\alpha)$? If the answer to Question 1 is "yes," the answer to Question 2 should be "yes" as well, since we'd have $V(\beta\alpha)=V(\alpha^{-1}\alpha\beta\alpha)=V(\alpha\beta)$.
Am I missing something here?
