Chord diagrams derived for n = 2,3,4 are given in the corresponding figure. Among all chord diagrams we can distinguish 2-vertex connected graphs (containing the edges of an 2n-gon as well), corresponding to non-prime KLs, and others, 3-connected, corresponding to prime KLs. 

For coloring of chord diagrams we have the rule: every two diagonals crossing each other must have different colors. A chord diagram will be colorable iff it is planar. The other, purely visual, criterion for colorability is the following: a chord diagram is colorable iff crossings of its diagonals do not form a polygon with an odd number of edges, and three or more diagonals have not a common point. Coloring of a (colorable) chord diagram represents a projection of a polyhedron enclosed in an 2n-gon, with proper visibility of all edges. 

In the case of 2-vertex connected chord diagrams, coloring is not unique: from the same uncolored chord diagram we can obtain several different colored diagrams. KL shadows, their corresponding self-avoiding curves and colored chord diagrams for n = 2,3,4 are given in the following figure, and for n = 5 in in the next figure. In the case of 3-vertex connected chord diagrams, a coloring is completely forced by the coloring of one edge: by choosing its color we can obtain only one colored chord diagram, or its dual. Hence, in the case of 3-vertex connected planar diagrams, an uncolored chord diagram provides a complete information about the corresponding self-avoiding curve. Every uncolored chord diagram can be given as a list of unordered pairs of numbers denoting chords. For example, the uncolored chord diagram from the figure (a) can be denoted as {{1,3},{2,6},{4,9},{5,8},{7,10}}. The same figure illustrates its bicoloring (a), the reconstruction of its corresponding self-avoiding curve (b-e), and KL shadow obtained (f). 

By restricting our attention to 3-vertex connected planar chord diagrams corresponding to prime KLs, for n = 2,3,...,8 we obtain, respectively, 1, 1, 3, 7, 33, 148, 923 chord diagrams corresponding to self-avoiding curves derived from prime KLs. For n = 2 we have one chord diagram {{1, 3},{2, 4}}, and for n = 3 one diagram {{1, 3}, {2, 5}, {4, 6}}. For n=3 there are three diagrams, given in the following table: