Prime Links

PRIME LINKS

One may recall that in the discussion of knots, we decided not to include in the tables the so called "connected sums", such as for instance {(1,4),(3,6),(5,2),(7,10),(9,12),(11,8)} (connected sum of two trefoils). This was done not only for "historical" reasons (i.e. everybody else does the same), but also for practical purposes. We thus ended up having only prime knots in the tables.

In a similar manner, we desire to include only "prime" links in the table of links. In contrast to knots, however, that can be divided into only two categories ("connected sums" and "prime"), the situation is not so simple with links. One may have for instance the "union" of unlinked pieces. As an example, one may consider the link that can be denoted as {(1,4),(3,6),(5,2),<7,14>,(8,11),(10,13),(12,9)}, which is just the "union" of two trefoils that are not linked to each other (i.e. their projections do not intersect). In addition, one may also have unions of connected sums.

In the case of connected sum of knots, if we are given the "connected components" and the order in which they are placed, one can uniquely determine the knot type they form. In the case of links, this happens only with unions (where one does not even need to know their order). When however one forms the connected sum of prime links, one also needs to know which component of each link is the one that is "connected".

We avoid these difficulties by excluding unions, connected sums and unions of connected sums from our tabulation. But how do we identify such cases? The simple technique with connected sums of knots (based on the idea of finding numbers i and j so that numbers between i and j are connected only with each other), does not always work for the links. If we use this method we may obtain the "unions", but not the "connected sums". This is because once we add the artificial crossings, the rule above may be violated, even if the link is not prime. We thus proceed as follows.

One may recall that after obtaining the signs of each crossing, we continued checking "drawability" by finding (and then counting) the areas the notation implies. Once we know which crossings are the artificial, we alter these areas accordingly, so that for each area we mark the segments that form its boundary. One may first notice that each crossing belongs to four areas and is the intersection of four segments. When we consider (and remove in this process) an artificial crossing, two of the areas are going to merge into one, while the four segments merge into two (see figure below).

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In the figure above, the top right and the bottom left areas are going to merge. In addition, the left and the top segments are going to merge, and so will the right and the bottom segments.

Once we know the segments of each area, we can easily check whether the link "looks" prime. Two areas are allowed to have one segment in common. But if they have a second common segment, then such a link "looks" a connected sum.

One thus establishes the following procedure. First, one attempts to find numbers i and j such that all numbers between i and j are paired to each other. If so, the link is definitely a union. If not, one obtains the segments of the various areas, according to the procedure presented above. If there do exist two areas with at least two segments in common, then the link is either a connected sum, or (if it is prime), it is not in the presentation with the least amount of crossings. In either case it can be discarded.

If a link presentation "survives" these two tests, the link may not necessarily be prime. In such a case one may need to perform appropriate equivalence moves in order to separate the presentation into its various pieces. In order to show that a link is actually prime, one needs to obtain the invariants of all combinations of connected sums, unions and unions of connected sums with crossings fewer than the crossings of the link under consideration, and show that they are different from this link's invariants. Finding these invariants is relatively simple, once we know the invariants of each piece.

If two links have A and B number of components, their union has A+B and their connected sum has A+B-1 components.

As far as linking numbers of unions are concerned: if two components belong to the same link, their linking number in the union does not change. If they belong to different links, their linking number in the union is 0.

The situation with linking numbers of connected sums is slightly more complicated. If component X of the first link is connected to component Y of the second, then X and Y merge to a component Z. This component now has linking numbers equal to the numbers that X and Y had in their original links. Any other two components: if they belonged to the same link, their linking number does not change. If not, their linking number is 0.

Colorings: if under a certain set of coloring rules one link accepts i distinct "colorizations" and the other accepts j such "colorizations", their union accepts the product of i times j "colorizations", and their connected sum accepts i times j divided by the total number of colors.

To continue with the rest of the discussion, and read about the link tabulation algorithm, please click here.


Charilaos Aneziris

Copyright 2001

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