= Kauffman bracket
{c}
{title2=$\langle D\rangle$}
{wiki}
For a nonempty unoriented <link diagram> $D$, the reduced Kauffman bracket is the <Laurent polynomial>
$$
\langle D\rangle=\sum_s A^{a(s)-b(s)}(-A^2-A^{-2})^{c(s)-1}.
$$
The 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.<State models and the Jones polynomial by Louis H. Kauffman>[Definition 2.1 and Figure 2, pages 396–397]{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 $-A^2-A^{-2}$.
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