The Theory of Links

An Introduction to the Theory of Links

During the past few years, we have presented an outline of the Theory of Knots, and in particular we have discussed the problem of Knot Tabulation. Readers interested in this discussion are referred to the relevant pages.

It turns out, however, that knots are merely a specific case of a more extended class of mathematical objects, which are usually referred to as links. While knots are formally defined as ambient isotopy classes of continuous one-to-one functions from the circle to the three-dimensional sphere, links are ambient isotopy classes of continuous one-to-one functions from a union of any number of circles to the three-dimensional sphere. This means that knots are a special example of links, where this "any number of circles" is equal to one.

In plain language: a knot may be indicated by drawing a closed line that does not intersect itself, while a link may be indicated by drawing a collection of closed lines, each of whom does not intersect itself, and no two such lines intersect each other.

Starting from the current page, we are going to extend the discussion from knots to links. We shall start by pointing out which aspects of the theory of knots are also applicable to links, and which aspects will now have to be modified. We are then going to continue with the problem of Tabulation of Links, where a number of new challenges emerges. Finally we are going to provide the reader with specific results.

To continue this discussion, please click here.

Charilaos Aneziris

Copyright 2001


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