 Kauffman bracket

For a nonempty unoriented link diagram $D$, the reduced Kauffman bracket is the Laurent polynomialThe sum runs over bracket smoothing states; $a(s)$ and $b(s)$ count the two smoothing choices, and $c(s)$ counts the resulting circles. Thus one circle has bracket one, each additional disjoint circle multiplies by $\delta =-A^2-A^{-2}$, and a crossing expands with weights $A$ and $A^{-1}$. The smoothing convention is fixed by marking the two sectors swept out when the overpassing line turns counterclockwise toward the underpassing line as the $A$ sectors.^[ref]

The Kauffman bracket is invariant under the second and third Reidemeister moves. A positive or negative curl instead multiplies it by $-A^3$ or $-A^{-3}$, respectively. Correcting this factor by the writhe of a link diagram produces the Jones polynomial. For two crossing-free circles, the bracket is simply $-A^2-A^{-2}$.