= Solution
Apply the given <skein relation> to a positive and a negative kink in a diagram of the <unknot>. Both crossed <link diagrams> represent the <unknot>, whereas the <oriented smoothing> represents the two-component <unlink>. Thus
$$
q-q^{-1}=(q^{1/2}-q^{-1/2})V(U_2).
$$
Factor the left-hand side and cancel the nonzero factor in the <Laurent polynomial> ring:
$$
\boxed{V(U_2)=q^{1/2}+q^{-1/2}.}
$$
The positive sign follows from the <skein relation> as printed in this paper. It differs from the frequently used <Jones polynomial> convention whose two-component <unlink> has value $-(q^{1/2}+q^{-1/2})$. With the printed convention, an $r$-component <link> has invariant $(-1)^{r-1}$ times that convention's <Jones polynomial>.
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