I've decided to learn some knot theory during this summer, using The Knot Book. Today, I showed that the Jones polynomial satisfies the Skein relation $$t^{-1}V(L_+) - tV(L_-) + (t^{-1/2}-t^{1/2})V(L_0)$$.
After showing this, I decided to work out an example. I picked the right-handed trefoil. We have previously calculated that the Jones polynomial of the riht-handed trefoil is $t+t^3-t^4$. All the crossings are negative, so I substituted the Jones polynomial of the trefoil for $V(L_-)$, and the switched the crossing to obtain $L_+$ and $L_0$. I obtained that $L_+$ can be turned into the unknot by a couple of Reidemeister moves, and hence decided that $V(L_+) = 1$. Finally, $L_0$ turns out to be the Hopf link, which has Jones polynomial $-t-t^{-1}$. So I substituted $V(L_0) = -t-t^{-1}$.
Then, looking at my Skein relation, I have:
$$t^{-1}(1)-t(t+t^3-t^4) + (t^{-1/2}-t^{1/2})(-t-t^{-1}) = t^{-1} - t^2 - t^4 + t^5 - t^{1/2} - t^{-3/2} + t^{3/2} + t^{-1/2} \neq 0$$
Why is this not working out correctly? What am I missing? I've checked that my polynomials for the trefoil and the Hopf link is correct, and I've checked my algebra, so I assume I'm missing something more fundamental.