Finite type invariant

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In the mathematical theory of knots, a finite type 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. It is then said to be of type m. These invariants were first studied by Mikhail Goussarov and Victor Vassiliev, independently, and so are often called Goussarov-Vassiliev invariants or Vassiliev invariants.

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 $V 1$ to be defined on a knot with one transverse singularity.

Consider a knot K to be a smooth embedding of a circle into $\mathbb R^3$ . Let K' be a smooth immersion of a circle into $\mathbb R^3$ with one transverse doube point. Then $V 1 (K') = V (K +) - V (K -)$ , where $K +$ 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.

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 similar notion of finite type invariant for 3-manifolds.

1 Some first invariant
2 Invariants representation
3 Order 4 invariant
4 Universal Vassiliev invariant
5 References

[edit] Some first invariant

The first nontrivial Vassiliev invariant of knots is of order 2. Moulo 2 it is equal to Arf invariant, and it can be obtained from the Conway polynomial - the $x^2\,$ coeficient .

[edit] Invariants representation

Polyak and Viro proved that all the Vassiliev invariants can be represented using chord diagrams. They gived description of first nontrivial invariant (of order 2 and 3) using such chord diagrams. not only order 2 and 3, but order 4 invariant was shown by Polyak and Viro, but later it was proven, that it isn't invariant at all - the formula given was wrong.

[edit] Order 4 invariant

after bad attempt by Polyak & Viro to write down order 4 invariant formula this work was made by Turaeva. nowadays it's proven that her formula was wrong too.

[edit] Universal Vassiliev invariant

Or, as it is more often named Kontcevich integral after Maxim Kontcevich. Kontcevich proved in 1993 the most important theorem about Vassiliev invariants. Existence of Kontcevich integral is equal to totality of Vassiliev invariants.

[edit] References

Mikhail Goussarov,
Victor A. Vassiliev, Cohomology of knot spaces. Theory of singularities and its applications, 23--69, Adv. Soviet Math., 1, Amer. Math. Soc., Providence, RI, 1990.
J. Birman and X-S Lin, Knot polynomials and Vassiliev's invariants. Invent. Math., 111, 225--270 (1993)
Dror Bar-Natan, On the Vassiliev knot invariants. Topology 34 (1995), 423--472