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  "body": "\n## Definition\n\nLet $D$ be an unoriented diagram of a tame [[link|1A5WRCNTM3]] in $\\R^3$, with crossing set $X$.\nAt a crossing, the *A-regions* are the two regions swept out when the over-strand is turned counterclockwise.\nThe *A-smoothing* joins the two A-regions, and the *B-smoothing* joins the other two regions.\nA *state* $s$ of $D$ chooses a smoothing at each crossing; let $a(s)$ and $b(s)$ count its A- and B-smoothings.\nLet $\\abs{s}$ be the number of circles in the resulting diagram without crossings.\nThe *Kauffman bracket* of $D$ is\n$$\\langle D \\rangle = \\sum_{s} A^{a(s) - b(s)} \\, (-A^2 - A^{-2})^{\\abs{s} - 1} \\in \\Z[A, A^{-1}],$$\nwhere the sum runs over all $2^{\\abs{X}}$ states.\n\n## Remarks\n\nEquivalently, $\\langle O \\rangle = 1$ 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.\nExample: a one-crossing diagram of the [[unknot|1TQ06R4Q66]] has one state with one circle and one with two circles.\nDepending on the sign of the kink, the bracket is $A + A^{-1}(-A^2 - A^{-2}) = -A^{-3}$ or its mirror value $-A^{3}$.\nNon-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.\nIt is invariant under the second and third [[Reidemeister moves|T3HAGWYKK7]], and the first move multiplies it by $-A^{\\pm 3}$.\nFor 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|4SPC17FPXE]].\n",
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