After introducing proper shadows, we can return to Kirkman's problem- systematic classification and enumeration of all minimized plane curves with n crossings. In solving this problem, Dowker and Gauss codes will be used. First we compute all distinct permutations of series of even numbers 2,4,...,2n. After producing all their partitions into 1,2,...,2n parts we obtain all possible Dowker codes of KL shadows with n crossings. Then we delete from them non-proper shadows according to Gauss codes. The result obtained are all possible potential Dowker codes of KL shadows with n crossings. We say "potential" because not all of them are necessarily realizable. For example, if you try to draw the potential Dowker code {{5},{8,10,2,4,6}} you will very soon come to a dead-end. If you test the planarity of the knot shadow graph obtained from it, you will find that it is not a planar graph. In fact, it is K5. It could happen also that some potential Dowker code is not-realizable (i.e., if you try to draw it, you will come to a dead-end), but its corresponding graph is planar. For example, to the non-realizable Dowker code {{6}, {4, 6, 8,10, 12, 2}} corresponds the (non-realizable) Gauss code {1,2, 3, 1, 4, 3, 5, 4, 6, 5, 2, 6}, and the planar graph 

{{1, 2}, {1, 3}, {1, 4}, {1, 6}, {2, 3}, {2, 5},{2, 6}, {3, 4}, {3, 5}, {4, 5}, {4, 6}, {5, 6}}
that is realizable as the link {{2, 2,2}, {6, 8, 10, 12, 2, 4}}. 

After that we can pose a strange question: where can certain non-realizable Dowker codes be realized? The answer is: on some surfaces other then plane Â2 (or sphere S2). The graph K5 corresponding to the code {{5},{8,10,2,4,6}} can be embedded on a torus, but there, applying the Component Algorithm we will not obtain a shadow of a knot, but a shadow of a 3-component link. One of its Gauss codes is {1,5,3},{2,5,4},{1,2,3,4}, but its Dowker code does not exist. 

The virtual knot theory introduced by L.Kauffman (1997, 1999, 2000) represents a "non-realizable" counterpart of the usual knot theory and gives the alternative answer to our question about realizability of Dowker codes.
 

 

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