Finite type invariant

In the mathematical theory of knots, a finite type invariant, or Vassiliev invariant, is a knot invariant that can be extended (in a precise manner to be described) to an invariant of certain singular knots that vanishes on singular knots with m + 1 singularities and does not vanish on some singular knot with 'm' singularities. It is then said to be of type or order m.

We give the combinatorial definition of finite type invariant due to Goussarov, and (independently) Joan Birman and Xiao-Song Lin. Let V be a knot invariant. Define V¹ to be defined on a knot with one transverse singularity.

Consider a knot K to be a smooth embedding of a circle into ${\displaystyle \mathbb {R} ^{3}}$. Let K' be a smooth immersion of a circle into ${\displaystyle \mathbb {R} ^{3}}$ with one transverse double point. Then ${\displaystyle V^{1}(K')=V(K_{+})-V(K_{-})}$, where ${\displaystyle K_{+}}$ is obtained from K by resolving the double point by pushing up one strand above the other, and K_- is obtained similarly by pushing the opposite strand above the other. We can do this for maps with two transverse double points, three transverse double points, etc., by using the above relation. For V to be of finite type means precisely that there must be a positive integer m such that V vanishes on maps with m + 1 transverse double points.

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