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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?

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Read the paper by Jones where he defines the polynomial. The relations are there. MR0766964 (86e:57006) Jones, Vaughan F. R. A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. – Ryan Budney Mar 4 '11 at 21:52
These relations hold for any trace invariant of braids, since trace is conjugation-invariant. – Ryan Budney Mar 4 '11 at 22:09

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