In the case of knots, as already discussed, the only possible way to alter a notation without affecting the regular presentation, was to move the starting point. In such case, every crossing number i would be replaced by (i+c)mod(2N), where c is a constant. Due to the fact, however, that we consider a knot equivalent both to its mirror symmetric and to its inverse (and to the mirror symmetric of its inverse), we also allow for changes where i is replaced by 2N+1-i, and where all pairs (i,j) are replaced by (j,i).
In the case of links, this group of notation changes is incomplete. One may alter the notation without affecting the regular presentation by changing the artificial crossings which were added during the notation process. Take for instance the notations A = {(1,4),(3,8),(5,10),(7,2),(9,12),<11,6>} and B = {(1,4),(3,8),<5,10>,(7,12),(9,2),(11,6)} (click here to see the figures ). (We use the convention that numbers inside () denote real crossings, while numbers inside <> denote artificial). These notations seem quite different. One may check that they cannot be transformed to each other by a mere change of the starting point, and they originate from different knot shadows.
Once more, one may easily "see" that they yield identical link presentations. But how can this be shown in a way independent of figures?
As shown in an earlier file, these two notations define the following "chains of neighbors":
One may see that it is possible to match the numbers of notations A and B as follows.
A1:B1 A2:B2 A3:B3 A4:B4 A5:B11 A12:B12 A7:B9 A8:B8 A9:B7 A10:B6
(This means that number 1 from notation A is matched to number 1 from notation B etc.)
This "match" has the following property. Let a number i from notation A that is paired to number j and has numbers k and l as its neighbors. Let now i, j, k, l be "matched" to numbers i', j', k', l' from notation B. Then, in notation B, i' is paired to j' and has k' and l' as its neighbors. In addition, all overcrossing numbers of A are matched to overcrossing numbers of B, and so do the undercrossing numbers.
If such a relationship exists between two notations, or if all overcrossings of A are matched to undercrossings of B, then and only then these two notations yield identical link presentations. One may notice for the special case of knots, where there is only one "chain of neighbors" 1-2-3-...-2N-1, this condition is equivalent to the one presented in the discussion of knots.
Now that these issues have been resolved, we must establish a procedure that will determine which among two link presentations is ordered "ahead" of the other. To go on with the discussion, please click here.
Charilaos Aneziris
Copyright 2001
Educational institutions are encouraged to reproduce and distribute these materials for educational use free of charge as long as credit and notification are provided. For any other purpose except educational, such as commercial etc, use of these materials is prohibited without prior written permission.