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.
For V to be of finite type means precisely that there must be a positive integer m such that V vanishes on maps with
The simplest nontrivial Vassiliev invariant of knots is given by the coefficient of the quadratic term of the Alexander–Conway polynomial.
[1] 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.
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.