$(1) \Rightarrow (2)$: Type-$n$ invariants define weight systems.

Already established: if $v$ is type $n$, then for singular knots with exactly $n$ double points, $v(K)$ depends only on the chord diagram $D(K)$ (since crossing changes in the non-singular part produce singular knots with $n+1$ points, on which $v$ vanishes). The 4T relation follows from considering a specific family of singular knots.

4T relation derivation. Consider a singular knot $K$ with $n-1$ double points and a region where two strands are close but do not intersect. By introducing one additional double point in four different ways (corresponding to four chord diagrams that differ in how the new chord is placed relative to the two strands), the four resulting singular knots satisfy a linear relation coming from the type-$n$ condition applied to a singular knot with $n+1$ points. Explicitly: the resolution of a triple-point singularity (three strands meeting) gives the 4T relation.

$(2) \Rightarrow (3)$: Trivial, since a function on chord diagrams satisfying 4T is precisely an element of $\mathcal{A}_n^*$.

$(3) \Rightarrow (1)$: Weight systems lift to Vassiliev invariants.

This is the deep direction, requiring the Kontsevich integral. Given $w \in \mathcal{A}_n^*$, define $v_w(K) = w(Z_n(K))$ where $Z_n$ is the degree-$n$ part of the Kontsevich integral. Then:

• $v_w$ is a knot invariant (since $Z$ is).
• For singular knots with $n+1$ points: $Z_n(K_\times)$ involves only chord diagrams from the resolutions, and the alternating sum over $>n$ double points produces zero in $\mathcal{A}_n$.
• Therefore $v_w$ is type $n$, and its associated weight system is $w$.

Exactness. We need to verify:

• $\phi$ is surjective: every weight system $w$ equals $\phi(v_w)$ by the Kontsevich construction.
• $\ker\phi = \mathcal{V}_{n-1}$: if $\phi(v) = 0$, then $v$ vanishes on all singular knots with $n$ double points, so $v$ is type $n-1$.

This completes the proof. $\square$