= Monodromy group of z squared plus z to the minus two
{title2=$z\mapsto z^2+z^{-2}$}
The degree-four <rational map> $F(z)=z^2+z^{-2}$ has branch values $-2,2,\infty$. Away from their inverse images, the transformations
$$
z\mapsto-z,
\qquad z\mapsto z^{-1}
$$
are commuting deck involutions. They generate a <Klein four-group> acting transitively on each fibre
$$
\{z,-z,z^{-1},-z^{-1}\}.
$$
Consequently the monodromy group is $V_4$, acting by the identity and the three double transpositions.
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