Finite type invariant

In the

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 ${\displaystyle K_{-}}$ 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 ${\displaystyle m+1}$ transverse double points.

Furthermore, note that there is notion of equivalence of knots with singularities being transverse double points and V should respect this equivalence. There is also a notion of finite type invariant for 3-manifolds.

Examples

The simplest nontrivial Vassiliev invariant of knots is given by the coefficient of the quadratic term of the

Any coefficient of the Kontsevich invariant is a finite type invariant.

The

Michael Polyak and Oleg Viro gave a description of the first nontrivial invariants of orders 2 and 3 by means of Gauss diagram representations. Mikhail N. Goussarov has proved that all Vassiliev invariants can be represented that way.

The universal Vassiliev invariant

In 1993,

• Willerton's fish

References