Vive la Difference!


As pointed out in the preceding page, the problem of the tabulation of links is quite different from the tabulation of knots. To be more specific, it is far more involved.

While the four challenges presented in the page Denoting Regular Knot Projections are exactly the same, the notation presented in that page is not applicable. In the case of knots, we would select some starting point, move along the knot, and associate each successive crossing point with a successive positive integer number, starting from 1 and ending with 2N (N is the crossing number of the regular presentation). We would continue with this process until we were back to the starting point. Then, we would denote the knot by a set whose elements would be the N ordered pairs of numbers: (undercrossing, overcrossing).

If we were to apply the same procedure for a link, we would end up having covered only one of the components of the link, and ignored all the others. Such a "notation" would indeed violate all conditions that the four challenges would impose. In order to avoid such a situation, we are now going to extend the knot notation, so that it can be applicable to links and pass successfully the four notation challenges. This extension is done as follows.

One starts with a regular link presentation. Then, using a process that resembles an inverse skein method, one adds (instead of removing) a number of "artificial" crossings, until the link becomes a knot. (One may show that if a link consists of n components, the number of such artificial crossings is at least n-1). Then, one denotes the resulting knot projection through the method introduced above. To denote the link projection, one replaces the pairs corresponding to the artificially added crossings, with unordered number pairs.

This notation methodology is going to become more clear once the reader goes through the following example. To continue with the rest of the discussion, please click here.

Charilaos Aneziris

Copyright 2001


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