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It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/link by changing the variable $V_L(t) \to V_L(t^{-1})$. More generally, the HOMFLY-PT polynomial of the mirror image can be obtained by switching from $P_L(l,m) \to P_L(l^{-1},m)$.

So can an equivalent statement be made about the Coloured Jones Polynomial of an arbitrary link, for the same colouring? For a link $L$ and mirror image $L^\star$ is the relation $C^{L}_{N,M,\cdots}(t) = C^{L^\star}_{N,M,\cdots}(t^{-1})$ true? Is there a reference for this?

This does seem to be the case in the paper by Habiro on pages 3 and 4, for the left and right handed trefoil knots as well as the left and right handed Whitehead Links

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    $\begingroup$ I am pretty sure this is true. If it is the proof would to be understanding how taking duals affects the highest weights of your modules, and this should correspond to the inversions you mentioned. However I haven’t thought about this carefully and I’m not sure if exactly the statement you want is written down anywhere. $\endgroup$
    – Calvin McPhail-Snyder
    Commented Feb 20 at 14:47
  • $\begingroup$ Actually you might need to be a little more careful: for a mirror image (not just orientation change) you also need to invert the R matrices and that’s a more complicated change. $\endgroup$
    – Calvin McPhail-Snyder
    Commented Feb 20 at 14:48
  • $\begingroup$ It should be easy to check with the help of the introduction by Murakami arxiv.org/abs/1002.0126 and in particular the explicit formula 2.2 for R and 2.4 for its inverse. $\endgroup$
    – Abdelmalek Abdesselam
    Commented Feb 20 at 21:26

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