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  "body": "\n## Definition\n\nLet $G$ be an abelian group and $v$ a [[knot invariant|GZQ5ZE3X6G]] of oriented [[knots|QYEAZTYGPT]] with values in $G$.\nA *singular knot* is an immersion of the oriented circle in $\\R^3$ whose only singularities are finitely many transverse double points.\nAt a double point, let $K_+$ and $K_-$ be its positive and negative resolutions into crossings.\nExtend $v$ to singular knots by the rule $v(K_\\times) = v(K_+) - v(K_-)$, applied at each double point.\nThe invariant $v$ is of *type* $\\le n$ if its extension vanishes on every singular knot with more than $n$ double points.\nIt is a *finite type invariant*, or *Vassiliev invariant*, if it is of type $\\le n$ for some $n$.\n\n## Remarks\n\nThe extension is well defined: resolving the double points in different orders gives the same alternating sum over all $2^k$ resolutions.\nThe invariants of type $\\le 0$ are the constants, and for knots so are those of type $\\le 1$.\nExample: the coefficient $c_2$ of $z^2$ in the Conway polynomial is of type $\\le 2$.\nNon-example: $c_2$ is not of type $\\le 1$, since it takes the value $1$ on a singular knot with two double points whose chords cross.\nAfter the substitution $t = e^h$, the coefficient of $h^n$ in the [[Jones polynomial|4SPC17FPXE]] is of type $\\le n$.\n",
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