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The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\Sigma_1)$, then $$Z(M_U) = Z(M_{-U})$$ where $-I\in \mathrm{Mod}(\Sigma_1)$ is the hyperelliptic involution.

What about the genus 2 case? If $-I\in \mathrm{Mod}(\Sigma_2)$ is the hyperelliptic involution on the genus 2 surface, then is there any $U\in \mathrm{Mod}(\Sigma_2)$, for which the following does not hold? $$Z(M_U) \stackrel{?}{=} Z(M_{-U})$$

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I’m not sure about the invariant due to the framing ambiguities. But for Witten-Reshetikhin-Turaev TQFT representations, I think it cannot detect the hyperelliptic involution in genus 1 and 2, since the basis for the TQFT vector space have the symmetry in the sense flipping by hyperelliptic involution gives the same basis vector back, hence in these two cases hyperelliptic involution acts trivially (projective). Which means also the invariant cannot detect it up to U(1).

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