The following theorem holds: if a diagram of a KL can be labelled with elements from a group G, then every diagram of the KL can be labelled with the elements from G, regardless of the choice of orientation (Livingston, 1993).
To prove this theorem it is sufficient to check what happens to the labelling after performing each Reidemeister move.
The use of labelling is one of the most powerful tools for distinguishing KLs. For example, it was very efficiently used by M. Thistlethwaite for the computer derivation of knots. In the compilation of knots with n £13 crossings, 12965 knots were represented by only 5639 different Alexander polynomials. Using labelling from all subgroups of S5, enebled him to reduce the number of unresolved cases to about a thousand (Thistlethwaite, 1985). Alexander polynomial can be considered as a special realization of the quandle (Manturov, 2004, page 57).
Elements g and g' in a group G are called conjugate if there is an element h Î G such that h-1gh = g'. The conjugacy is an equivalence relation, that preserves the cyclic structure of a permutation group G: every element g and all its conjugates are represented as the products of permutation cycles in the same way. The relation of conjugacy induces partitions of G into equivalence classes. For example, in S5 there are seven conjugacy classes, that can be represented by
If a diagram of an oriented knot can be labelled with elements of a group G with the labels coming from a conjugacy class C of G, then every diagram of the same knot can be labelled with the elements from C. In the case of links, labels on each component of a labelled link belong to the same conjugacy class.
Using consistency relationships, once a few labels are chosen, the rest are forced. In practice, we take a knot K and fix a group G that will be used for knot labelling. After labelling two knot generators in some vertex by two group generators x and y, the third generator in that vertex will be labelled by the consistency condition. Hence, each crossing determines a label on the next arc, forced by the labels that preceded it. The labels of some arcs will be forced by consistency conditions coming from their two ends. Equations satisfied in G can be read from the labelling of arcs. In this way, the knot labelling problem is reduced to solving the equations obtained from the group G.
C. Livingston (1993) proposed an interesting example of knots 4 2 (61) and 3,3,-3 (946), that cannot be distinguished using colorings, or by their Alexander polynomials, but are distinguishable by a suitably chosen labelling.
Let us consider knot 4 2 and its labelling using transpositions from S4. By the choice of two labels x and y in any crossing of the knot 4 2, all the other labels are forced by the consistency relation. Since only two transpositions from S4 are not enough to generate S4, we conclude that it is impossible to construct a labelling of knot 4 2 using transpositions from S4. The labelling of the knot 3,3,-3 by using transpositions from S4 proves that the knots in question are different.