Finite type invariant

Definition

Let $G$ be an abelian group and $v$ a knot invariant of oriented knots with values in $G$ . A singular knot is an immersion of the oriented circle in $\R^3$ whose only singularities are finitely many transverse double points. At a double point, let $K_+$ and $K_-$ be its positive and negative resolutions into crossings. Extend $v$ to singular knots by the rule $v(K_\times) = v(K_+) - v(K_-)$ , applied at each double point. The invariant $v$ is of type $\le n$ if its extension vanishes on every singular knot with more than $n$ double points. It is a finite type invariant, or Vassiliev invariant, if it is of type $\le n$ for some $n$ .

Remarks

The extension is well defined: resolving the double points in different orders gives the same alternating sum over all $2^k$ resolutions. The invariants of type $\le 0$ are the constants, and for knots so are those of type $\le 1$ . Example: the coefficient $c_2$ of $z^2$ in the Conway polynomial is of type $\le 2$ . Non-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. After the substitution $t = e^h$ , the coefficient of $h^n$ in the Jones polynomial is of type $\le n$ .

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Knot concept
Knot invariant concept
Finite type invariant definition

Tag	`FMWGQ0W2X0`
URI	`m2://FMWGQ0W2X0`
Kind	definition
Writing	W0 stub
MSC	57K16, 57K10
Defines	Finite type invariant
Uses	Knot, Knot invariant
Changed	2 November 2026

Finite type invariant

Definition

Remarks

Reading path

Further reading

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