For n = 5, the seven chord diagrams are given in the following table:
For n = 6, thirty three such diagrams and their corresponding self-avoiding curves given by mirror placements are illustrated in the corresponding figure. Different shadows of the same KL can give different self-avoiding curves, as in the case of the link 2 2 2.
Visual recognition of self-avoiding curves, either direct or from shapes, is complicated even for a small number of mirrors, but it is almost immediate from chord diagrams.
It is interesting to mention a possible connection between shapes originating from self-avoiding curves and some biological forms.
From every KL shadow can be derived one or several self-avoiding curves. Some conclusions about original KLs can be made based on the chord diagrams of those self-avoiding curves. For example, to every diagonal connecting two vertices separated by one vertex, and to every pair of parallel adjacent diagonals corresponds a digon in the original KL shadow; diagrams without them correspond to basic polyhedra. This way, we can follow a process of bigon collapsing directly in chord diagrams.
Among all chord diagrams, we can distinguish antisymmetrical diagrams, that remain unchanged by opposite coloring of chords. With regard to self-avoiding curves, this means that the external region is equal to the internal one. Such diagrams are self-dual. For example, for n = 6, eleven among 33 chord diagrams are self-dual.
Again, families of KLs play important role as before, followed by families of chord diagrams and self-avoiding curves derived from them. Chord diagrams belonging to a same family can be visually recognized.
Self-avoiding curves can be embedded on different surfaces, so together with Viae Globi on a sphere S3, we can consider Viae Tori on a torus (later introduced in analogy to Sequin's Viae Globi), or on any other surface.
The LinKnot function fDiffViae (webMathematica fDiffViae) for a given number n derives all different self-avoiding curves with n mirrors that can be obtained from prime KLs with n crossings. For every such curve given by its (uncolored) chord diagram we can find its basic prime KL by using the LinKnot function fViaToKL (webMathematica fViaToKL).