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  "body": "\n## Definition\n\nIdentify $\\R^3$ with $\\C \\times \\R$, with complex coordinate $z$ and height $t$.\nLet $K$ be an oriented [[knot|QYEAZTYGPT]] in $\\C \\times \\R$ such that the height $t$ restricted to $K$ is a Morse function.\nLet $t_{\\min}$ and $t_{\\max}$ be the minimum and the maximum of $t$ on $K$.\nLet $\\mathcal{A}$ be the $\\C$-vector space of chord diagrams on the circle modulo the four-term relation and the one-term relation.\nThe one-term relation kills every diagram with an isolated chord.\nLet $\\widehat{\\mathcal{A}}$ be the completion of $\\mathcal{A}$ by degree.\nFor $t_{\\min} \u003c t_1 \u003c \\cdots \u003c t_m \u003c t_{\\max}$, let $P = \\set{(z_j, z_j')}_{j=1}^{m}$ be a choice of two distinct points of $K$ at height $t_j$, for each $j$.\nLet $D_P$ be the chord diagram with these $m$ chords, and $\\downarrow_P$ the number of chosen points where $K$ is oriented downward.\nThe *Kontsevich integral* of $K$ is\n$$Z(K) = \\sum_{m \\ge 0} \\frac{1}{(2\\pi i)^m} \\int_{t_{\\min} \u003c t_1 \u003c \\cdots \u003c t_m \u003c t_{\\max}} \\sum_{P} (-1)^{\\downarrow_P} D_P \\bigwedge_{j=1}^{m} \\frac{dz_j - dz_j'}{z_j - z_j'} \\in \\widehat{\\mathcal{A}}.$$\n\n## Remarks\n\nEach integral converges, and $Z(K)$ is unchanged by isotopies that keep the number of critical points of the height fixed.\nLet $c$ be the number of critical points and $H$ the unknot embedded with two maxima and two minima.\nThen the normalized $\\widehat{Z}(K) = Z(H)^{-c/2} Z(K)$, with the product in $\\widehat{\\mathcal{A}}$, is an isotopy invariant of $K$.\nExample: the degree-$0$ part of $Z(K)$ is the empty chord diagram with coefficient $1$, since the term $m = 0$ is the constant $1$.\nNon-example: the unnormalized $Z$ is not an invariant, because adding a pair of critical points multiplies it by $Z(H)$, which is not $1$.\nEvery rational-valued [[finite type invariant|TWH3WVKFF1]] of type $\\le n$ factors through $\\widehat{Z}$ followed by a linear map on the part of degree $\\le n$.\n",
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