A singular knot is an immersion $S^1 \to S^3$ with finitely many transverse double points (self-intersections). Let $\mathcal{K}_m$ denote singular knots with exactly $m$ double points. Any knot invariant $v: \{\text{knots}\} \to \mathbb{Q}$ extends to singular knots via the Vassiliev skein relation: at each double point, $v(K_\times) = v(K_+) - v(K_-)$ where $K_\times$ is the singular knot, $K_+$ is the positive resolution, and $K_-$ is the negative resolution. By iterating, $v$ extends to $\mathcal{K}_m$ for all $m$.

A knot invariant $v$ is a Vassiliev invariant of type $n$ (or finite-type invariant of order $n$) if $v(K) = 0$ for every singular knot $K$ with more than $n$ double points (i.e., $v|_{\mathcal{K}_{n+1}} = 0$). Equivalently, the Vassiliev extension of $v$ to singular knots with $n+1$ or more singularities vanishes identically. Let $\mathcal{V}_n$ denote the vector space of type-$n$ Vassiliev invariants. The filtration is $\mathcal{V}_0 \subset \mathcal{V}_1 \subset \mathcal{V}_2 \subset \cdots$ with $\mathcal{V}_0 = \mathbb{Q}$ (constants).

The associated graded space $\mathrm{gr}_n\mathcal{V} = \mathcal{V}_n/\mathcal{V}_{n-1}$ captures the "purely order-$n$" invariants. A type-$n$ invariant $v$ defines a weight system $w_v: \mathcal{K}_n/\mathcal{K}_{n+1} \to \mathbb{Q}$ (its top-order part). The dimensions $d_n = \dim(\mathcal{V}_n/\mathcal{V}_{n-1})$ for small $n$ are: $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, $d_3 = 1$, $d_4 = 3$, $d_5 = 4$, $d_6 = 9$, $d_7 = 14$, $d_8 = 27$, ... The total dimension grows exponentially; the exact growth rate is unknown.