The relation "over-under" can be included in a Gauss code by using colored numbers. In a Gauss code of a KL every number appears twice, and now it will appear once as white, and once yelow. In the case of an alternating KL, we will have an alternating sequence "over-under" (or "under-over") for each component. For example, to the trefoil knot diagram corresponds the Gauss code {1,2,3,1,2,3} or {1,2,3,1,2,3}, where one of the alternating KL diagrams is always a mirror image of the other in a mirror reflection plane coinciding with the projection plane Â2 . A KL is achiral (or amphicheiral) if it coincides with its mirror image. Otherwise, it is chiral. Trefoil knot is an example of chiral knot, and the figure-eight knot and Borromean rings are achiral. For Borromean rings, we have a Gauss code {{1,2,4,5},{3,1,6,4},{2,6,5,3}} or {{1,2,4,5},{3,1,6,4},{2,6,5,3}}. In translating them to Dowker codes, for a trefoil knot we obtain from the first Gauss code the list of ordered pairs ({1,4},{2,5},{3,6}}, the sorted list  D = {{1,4},{3,6},{5,2}}, and the corresponding Dowker code {{3},{4,6,2}} and from the second Gauss code the list of ordered pairs ({1,4},{2,5},{3,6}}, the sorted list D = {{1,4},{3,6},{5,2}} and the Dowker code {{3},{4,6,2}}. In the same way, from the Gauss codes of Borromean rings we obtain the first sorted list divided into components 

D={{{1,6},{3,8},{},{{5,12},{7,10}},{{9,2},{11,4}}}

and the second divided sorted list 

D={{{1,6},{3,8}},{5,12},{7,10}},{{9,2},{11,4}}}

so their corresponding Dowker codes are 

{{2,2,2},{6,8,12,10,2,4}} and {{2,2,2},{6,8,12,10,2,4}}, respectively. Let us notice that for alternating diagrams all Dowker codes are "lower" or "upper", meaning that all the numbers in the code are underlined or overlined, in contrast to non-alternating diagrams when some of them will be underlined, and others overlined. Two KL projections L' and L" are sense-preserving isomorphic iff there is an isomorphism of their corresponding graphs preserving the relation "over-under". If we are not interested in the handedness of the projections ("left" or "right"), the isomorphism mentioned can preserve or reverse all overcrossings into undercrossings and vice versa, transforming that way a KL projection into its isomorphic mirror image. In this case such projections will be called isomorphic. Otherwise, they are called non-isomorphic


 

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