Kontsevich integral

Definition

Identify $\R^3$ with $\C \times \R$, with complex coordinate $z$ and height $t$. Let $K$ be an oriented knot in $\C \times \R$ such that the height $t$ restricted to $K$ is a Morse function. Let $t_{\min}$ and $t_{\max}$ be the minimum and the maximum of $t$ on $K$. Let $\mathcal{A}$ be the $\C$-vector space of chord diagrams on the circle modulo the four-term relation and the one-term relation. The one-term relation kills every diagram with an isolated chord. Let $\widehat{\mathcal{A}}$ be the completion of $\mathcal{A}$ by degree. For $t_{\min} < t_1 < \cdots < t_m < t_{\max}$, 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 $j$. Let $D_P$ be the chord diagram with these $m$ chords, and $\downarrow_P$ the number of chosen points where $K$ is oriented downward. The Kontsevich integral of $K$ is $$Z(K) = \sum_{m \ge 0} \frac{1}{(2\pi i)^m} \int_{t_{\min} < t_1 < \cdots < t_m < t_{\max}} \sum_{P} (-1)^{\downarrow_P} D_P \bigwedge_{j=1}^{m} \frac{dz_j - dz_j'}{z_j - z_j'} \in \widehat{\mathcal{A}}.$$