Step 1: Setup. Consider a singular knot $K_0$ with $n-1$ double points (fixed) and a region where two strands $\alpha$ and $\beta$ of $K_0$ are close together without intersecting. We label four points on the two strands: $a_1, a_2$ on strand $\alpha$ (in order of orientation) and $b_1, b_2$ on strand $\beta$.

Step 2: Introduce a double point. There are four ways to add one more double point by pushing strands $\alpha$ and $\beta$ together:

• $D_1$: connect $a_1$ to $b_1$ (create a double point between the beginnings of the two strand segments).
• $D_2$: connect $a_1$ to $b_2$ (beginning of $\alpha$ with end of $\beta$).
• $D_3$: connect $a_2$ to $b_2$ (ends of both strands).
• $D_4$: connect $a_2$ to $b_1$ (end of $\alpha$ with beginning of $\beta$).

Each gives a singular knot with $n$ double points, and $v$ evaluates to a number on each.

Step 3: The key isotopy. Consider a singular knot $K_{n+1}$ with $n+1$ double points: the $n-1$ original double points of $K_0$, plus two additional double points between strands $\alpha$ and $\beta$. Since $v$ is type $n$, we have $v(K_{n+1}) = 0$ for any such configuration.

Now consider the specific double-point configuration where both new singularities occur between the same pair of strands. The Vassiliev skein relation at each of the two new double points produces a signed sum. Resolving the first double point: $0 = v(K_{n+1}) = v(K^+_{1}) - v(K^-_{1})$ where $K^+_1$ and $K^-_1$ are singular knots with $n$ double points.

Step 4: Detailed resolution. Consider a singular knot where strand $\alpha$ crosses strand $\beta$ twice (creating two double points between them, in addition to the $n-1$ existing ones). This $(n+1)$-fold singular knot can be created in different ways, depending on the relative positions.

The isotopy argument proceeds as follows. Slide the strand $\beta$ across the strand $\alpha$ along a small arc. This motion passes through exactly two singular configurations (where $\beta$ touches $\alpha$), creating two different singular knots with $n$ double points. The signed sum from the Vassiliev extension at these intermediate singular knots gives:

$v(D_1) - v(D_2) + v(D_3) - v(D_4) = v(K_{\text{before}}) - v(K_{\text{after}})$

Step 5: The move is an isotopy. The key observation: the overall motion of strand $\beta$ past strand $\alpha$ (starting configuration to ending configuration) is an ambient isotopy of the singular knot $K_0$ (with its $n-1$ fixed double points). Therefore $v(K_{\text{before}}) = v(K_{\text{after}})$ (since $v$ is a knot invariant on the level of the $n-1$ original singularities).

Step 6: Conclude. Combining Steps 4 and 5: $v(D_1) - v(D_2) + v(D_3) - v(D_4) = 0$

Since this holds for every type-$n$ Vassiliev invariant, the relation $D_1 - D_2 + D_3 - D_4 = 0$ holds in $\mathcal{A}_n$ (independently of the choice of $v$). $\square$