= Solution
Let $H_+$ be the positively oriented <Hopf link> obtained by closing a two-strand braid with two positive crossings. Switching one crossing gives the two-component <unlink>; the <oriented smoothing> gives the <unknot>. The printed <skein relation> gives
$$
qV(U_2)-q^{-1}V(H_+)=q^{1/2}-q^{-1/2},
\qquad V(H_+)=q^{5/2}+q^{1/2}.
$$
Represent the right-handed <trefoil knot> by the closure of the two-strand braid with three positive crossings. Switching one crossing gives the <unknot>, while its <oriented smoothing> gives $H_+$. Consequently
$$
q-q^{-1}V(T_+)=(q^{1/2}-q^{-1/2})(q^{5/2}+q^{1/2}),
$$
so the two <Jones polynomials> are
$$
\boxed{V(T_+)=q+q^3-q^4,\qquad V(T_-)=q^{-1}+q^{-3}-q^{-4}.}
$$
The second expression follows from the <mirror of a link> identity. They are unequal <Laurent polynomials>, so invariance under <link isotopy> proves that \b[the right-handed and left-handed trefoils are not isotopic].
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