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 3^{n}, 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 n^{th}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 oneterm 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 nonsingular 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 D_{n} of order n (i.e., with n double points). Then we can rewrite the oneterm relation and fourterm relation in the language of chord diagrams. By using the oneterm relation, we can reduce the list D_{n} to one diagram for n = 2 and two diagrams for n = 3, by eliminating the diagrams d_{11}, d_{22}, d_{34}, d_{35} and reducing d_{33} to d_{21}. If we are treating Gauss diagrams as vectors, according to the fourterm relation we obtain the relation d_{31} = 2d_{32}, so dim(D_{1}) = 0, dim(D_{2}) = 1, dim(D_{3}) = 1, etc. Recalling that dim(D_{0}) = 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 D_{n} for n = 0,1,2,...,9 (BarNatan, 1995). The space of chord diagrams with n chords modulo the oneterm and fourterm relations we denote by A_{n}. 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 V_{n}/(V_{n}1) of n^{th}order Vassiliev invariants is isomorphic to the space A_{n}. As a result, for every n we can obtain the actuality table A_{n}: 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. The function Vassiliev (webMathematica Vassiliev) from the program KnotTheory computes Vassiliev invariants of KL.
