Finite Type Invariant - The Universal Vassiliev Invariant

The Universal Vassiliev Invariant

In 1993, Maxim Kontsevich proved the following important theorem about Vassiliev invariants: For every knot one can compute an integral, now called the Kontsevich integral, which is a universal Vassiliev invariant, meaning that every Vassiliev invariant can be obtained from it by an appropriate evaluation. It is not known at present whether the Kontsevich integral, or the totality of Vassiliev invariants, is a complete knot invariant. Computation of the Kontsevich integral, which has values in an algebra of chord diagrams, turns out to be rather difficult and has been done only for a few classes of knots up to now. There is no finite-type invariant of degree less than 11 which distinguishes mutant knots.