For a knot $K$ embedded in $\mathbb{R}^3$ with coordinates $(z,\bar{z},t)$ where $t$ is a Morse function on $K$ (finitely many critical points), the Kontsevich integral is: $Z(K) = \sum_{m=0}^\infty \frac{1}{(2\pi i)^m}\int_{t_\mathrm{min} < t_1 < \cdots < t_m < t_\mathrm{max}}\sum_{P}\,(-1)^{\downarrow_P}\,D_P\,\bigwedge_{j=1}^m \frac{dz_j - dz_j'}{z_j - z_j'}$ where: the outer sum is over the number of chords $m$; the integral is over the simplex of increasing $t$-values; for each $t_j$, we choose a pair of points $(z_j,t_j)$, $(z_j',t_j)$ on $K$ at height $t_j$ (the chord endpoints); $P$ denotes a pairing of these points; $(-1)^{\downarrow_P}$ counts the number of downward-oriented strands among the selected points; and $D_P$ is the chord diagram recording the pairing.

The Kontsevich integral satisfies:

1. Well-defined: $Z(K)$ is invariant under deformations of $K$ that preserve the Morse function structure, and after a normalization (dividing by $Z(\text{unknot with same number of critical points})$), it is a knot invariant.
2. Group-like: $Z(K)$ is a group-like element of the completed Hopf algebra $\hat{\mathcal{A}}$, meaning $\Delta Z(K) = Z(K) \otimes Z(K)$.
3. Universality: For any weight system $w$ of degree $n$, the composition $w \circ Z_n$ is a Vassiliev invariant of type $n$. Moreover, every Vassiliev invariant arises this way. Thus $Z$ is the universal Vassiliev invariant.
4. Multiplicativity: $Z(K_1\#K_2) = Z(K_1)\cdot Z(K_2)$ (product in $\hat{\mathcal{A}}$).

The normalization of the Kontsevich integral requires the Drinfeld associator $\Phi_\mathrm{KZ} \in \hat{\mathcal{A}}(\uparrow\uparrow\uparrow)$ (a formal series of chord diagrams on three strands), which encodes the monodromy of the KZ equation. The associator satisfies the pentagon and hexagon equations, making it a key object in quantum group theory. The normalized (framing-independent) Kontsevich integral is $\hat{Z}(K) = Z(K)/Z(U_n)$ where $U_n$ is a standard unknot with $n$ maxima. Different choices of associator (related by gauge transformations) give the same final invariant.