Kauffman bracket

Remarks

Equivalently, $\langle O \rangle = 1$ for a single circle, adding a disjoint circle multiplies by $-A^2 - A^{-2}$, and each crossing expands as $A$ times its A-smoothing plus $A^{-1}$ times its B-smoothing. Example: a one-crossing diagram of the unknot has one state with one circle and one with two circles. Depending on the sign of the kink, the bracket is $A + A^{-1}(-A^2 - A^{-2}) = -A^{-3}$ or its mirror value $-A^{3}$. Non-example of invariance: this kinked diagram and the round circle depict the same knot but have brackets $-A^{\mp 3}$ and $1$, so $\langle \cdot \rangle$ is not a link invariant. It is invariant under the second and third Reidemeister moves, and the first move multiplies it by $-A^{\pm 3}$. For an oriented diagram with writhe $w(D)$, the product $(-A^3)^{-w(D)} \langle D \rangle$ is an invariant, and $A = t^{-1/4}$ turns it into the Jones polynomial.