I am trying to show that if we define a new invariant of knots $W(L)$ which follows the same rules as the Jones polynomial $V(L)$, so that it has value 1 on the unknot and satisfies the Skein relation $$t^{-1}W(L_+) - tW(L_-) + (t^{-1/2}-t^{1/2})W(L_0)=0$$ then we must have $W(L)= V(L)$.
I know some things which I think may come in handy; if we set $t=1$ in the Skein relation, we get $W(L_+)=W(L_-)$. We can change any link diagram to a diagram of trivial links by changing crossings from over to under and vice versa. And the Jones polynomial of a union of $k$ trivial links is $(t^{-1/2}-t^{1/2})^{k-1}$. I feel I should be able to put these together to get the result, but I'm not sure how.