If we agree to replace every pair of overlined numbers by the same numbers without overlinings, to replace every pair of underlined numbers by those numbers in opposite (descending) order, and sort the obtained list, the result is the list {{1,4},{5,8},{6,3},{9,12},{10,7},{11,2}}. The list of second elements in each pair gives the short code {4,8,3,12,7,2}. The complete code can be easily restored from the short code, knowing that the first parts of the ordered pairs are ordered numbers from the set {1,2,...,12}, not belonging to the set {4,8,3,12,7,2}, i.e., the numbers {1,5,6,9,10,11}. A different agreement: replacing every pair of overlined numbers by those numbers in opposite (descending) order, every pair of underlined numbers by the same pair of numbers and sort the obtained list, gives the dual list {{2,11},{3,6},{4,1},{7,10},{8,5},{12,9}}, and the dual short code {11,6,1,10,5,9}. 

Every self-avoiding curve can be graphically interpreted by a chord diagram: a regular 2n-gon, where points belonging to internal mirrors are connected by full, and points belonging to external mirrors by broken diagonal lines (or by black and white lines). For example, the first self-avoiding curve from the figure will be described by the chord diagram from the figure, or by its dual obtained by inverse bicoloring, where full (black) lines are replaced by broken (white) lines and vice versa. In the both cases chord diagrams are equal to their duals, i.e., they are self-dual. From every chord diagram we can easily obtain a code of the corresponding self-avoiding curve and vice versa, and to recover an original KL from which that curve is derived. This is illustrated in the following figure, where two stages of the recovering are shown. 

In order to derive and enumerate all self-avoiding curves with n mirrors, one should first derive all different non-colored chord diagrams, and then impose the appropriate coloring. The following rule holds for non-colored chord diagrams: every vertex belongs to exactly one diagonal (chord). In fact, we are searching for all different minimal sets of diagonals that span a regular 2n-gon, where sets that can be obtained one from another by symmetries of 2n-gon are considered to be the same. For n = 2,3,...,7 we obtain, respectively 1, 2, 7, 29, 176, 1788 such different sets. The sequence obtained is A003437 from the On-line encyclopedia of integer sequences, that represents the number of unlabeled Hamiltonian circuits on n-octahedron (Singmaster, 1975). An n-octahedron is the complete n-partite graph K_2,2,...,2 (n pairs of opposite vertices with edges connecting each vertex to every other vertex except its opposite). Singmaster notes that such a Hamiltonian cycle can be viewed as a way of seating n couples around a circular table so that no man is next to his wife. The number of cases is given by the formula (Pratt, 1996)