Analogous to the way we considered states of a KL shadow with regard to overcrossings and undercrossings, we can deal with special projections by giving three possible choices for every vertex (overcrossing, undercrossing, or double vertex). For a KL shadow with n vertices, the number of possible variations is 3n, but this can be considerably reduced thanks to the symmetry of a KL in question. We can consider Vassiliev invariants in terms of purely geometrical combinatorial theory by introducing Gauss diagrams (or chord diagrams) of the order n (n = 1,2,3,...). We already concluded that the value of an nth-order invariant of a KL with n double points is unaffected by crossing changes. This value is independent of the phenomenon of knotting, and depends only on the order in which double points appear when tracing KL.
In the case of knots we can introduce the following rule. Let a knot K with n double points be given. Proceeding around a circle we will label all double points visited by travelling in our oriented knot K in that order. Then we join each pair of points having the same label by a chord. The resulting diagram is called a Gauss diagram (or chord diagram) of K. The Gauss diagram of the special projection of a trefoil knot with three double points and one loop is given in the corresponding figure. In the same way as before, thanks to the one-term relation, we can delete all double points with a loop and continue to work with proper (or reduced) special projections and corresponding proper (or reduced) Gauss diagrams. All the non-singular knots have the same diagram- a circle without any cords. In general, many knots correspond to the same diagram. Now we can construct all different Gauss diagrams Dn of order n (i.e., with n double points). Then we can rewrite the one-term relation and four-term relation in the language of chord diagrams. By using the one-term relation, we can reduce the list Dn to one diagram for n = 2 and two diagrams for n = 3, by eliminating the diagrams d11, d22, d34, d35 and reducing d33 to d21. If we are treating Gauss diagrams as vectors, according to the four-term relation we obtain the relation d31 = 2d32, so dim(D1) = 0, dim(D2) = 1, dim(D3) = 1, etc. Recalling that dim(D0) = 1, and continuing the calculation for n = 3,4,...,9, we obtain the sequence 1, 0, 1, 1, 3, 4, 9, 14, 27, 44 of the dimensions of the spaces Dn for n = 0,1,2,...,9 (Bar-Natan, 1995). The space of chord diagrams with n chords modulo the one-term and four-term relations we denote by An. The main result of the combinatorial theory of Gauss diagrams and Vassiliev invariants is expressed by Kontsevich's theorem:
Kontsevich's Theorem The vector space Vn/(Vn-1) of nth-order Vassiliev invariants is isomorphic to the space An.
As a result, for every n we can obtain the actuality table An: the list of all independent Gauss diagrams with n chords. As the value of n increases, the calculation of actuality tables demands an enormous computing power.