As one may read in Prasolov and Sossinsky's text on knots, the Kauffman bracket assigns to each unoriented link diagram a Laurent polynomial, and one sees that it is invariant under the second and third Redemeister moves, but not the first. However, if we evaluate the Kauffman bracket at a primitive sixth root of unity we get a complex valued assignment which is fully invariant under all three moves, and is therefore an isotopy invariant.
Obviously this invariant is not at all powerful, since it even fails to distinguish trivial links with different number of components. I'm still interested in knowing if this invariant has a name and if it is indeed able to distinguish some important links.
The Kauffman bracket at a primitive sixth root of unity is the same as the Jones polynomial at $e^{2\pi i/3}$. It is known that no matter the link $L$, $V_L(e^{2\pi i/3})=1$. This is not too difficult to show with the skein relation for the Jones polynomial.