A Matter of Priority


Once we identify equivalent links, we need some criterion in each to order them. This criterion should be unambiguous, in the sense that for every two links L1 and L2, either L1 is ordered before L2, or L2 is ordered before L1. In addition, if link L1 is ordered before some link L2 and L2 before L3, then L1 has to be ordered before L3. There is one more condition we need in order to be able to start obtaining the list of links: each link should be preceded by a finite number of links.

Similarly to the case of knots, in order to establish an order for links, we do first need to establish an order for notations. Each presentation is ordered according to its "leading" notation, and each link is ordered according to its "leading" presentation. In a manner similar to knots, if a notation A consists of a total of Na crossings, a notation B consists of Nb crossings and Na < Nb, then A is ordered ahead of B. If Na = Nb, we check in each notation the "partner" of 1. The one with the smallest such partner is considered to be ordered ahead of the other. If these partners are equal, we consider the partners of 3. If they too are equal, we consider the partners of 3, 5, 7, ....

If the issue is still unresolved, unlike the case of knots, we establish now a different criterion. If in notation A crossing number 1 corresponds to a real crossing while in notation B it corresponds to an artificial one, then A is ordered ahead of B. If in both cases the crossings are either both real or both artificial, we consider the case with crossing number 3. If there is still no answer, we then consider crossing number 5, followed by 7, 9, 11, ....

If the issue is still unresolved, we apply now the criterion used in the case of knots: we check in these two notations whether crossing number 1 is overcrossing or undercrossing. A is ordered ahead of B if 1 is undercrossing in A and overcrossing in B. If necessary, we consider in a similar manner the case with crossing numbers 3, 5, 7, .... Naturally, if the issue is not resolved at this point, it would mean that A and B are identical.

While this criterion is adequate for our purpose, it leads to some "peculiar" results. To be more specific, the table of links is going to list successively the links that consist of 0, 1, 2, 3, 4, 5, ... crossings, which might be expected. But since this number is the sum of both real and artificial crossings, there are going to be cases where links with fewer real crossings are listed after links with more real crossings. The first such case is the order of the Hopf link and the trefoil. While the Hopf link consists of two real crossings and the trefoil of three, the trefoil is ordered ahead of the Hopf link. Another somewhat peculiar effect is the "mix up" between links of various number of components. As we shall see later when we show the actual results, for N = 5 we first obtain a one component link (knot), followed by a two component link, which is then followed by another knot. These "peculiarities", while not exactly desirable, are the smallest possible price one must pay, in order to be able to obtain a long table in a reasonable amount of time.

Please click here to continue with the discussion, where we introduce a class of "new" link invariants..

Charilaos Aneziris

Copyright 2001


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