The relation "overunder" 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 "overunder" (or "underover") 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 Â^{2} . 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 figureeight 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 listD={{{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 nonalternating diagrams when some of them will be underlined, and others overlined. Two KL projections L' and L" are sensepreserving isomorphic iff there is an isomorphism of their corresponding graphs preserving the relation "overunder". 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 nonisomorphic.
