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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

The Borromean and Hopf link are supposedly achiral, and switching to the inverse given the equations for CJP keeps it unchanged. But then there is this paper where it seems that this should only be the case for the Borromean link, not the Hopf link, so something seems to be going wrong

This question has been reposted from MathStackExchange

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