What do Knots and Links have in common

What do KNOTS and LINKS have in common?

When we were discussing the Theory of Knots, we had divided the material into two parts. In the first part, A Knot Theory Primer, we presented some basic elements of Knot Theory, which would be useful for our purpose. In the second part, The Problem of Knot Classofication, we discussed methods and techniques that are needed in order to create a list of topologically distinct knots.

It turns out that when one extends the discussion from knots to links, almost nothing of the first part is going to change, while on the other side, almost everything of the second part needs to be modified. To be more specific:

In the discussion presented in the ten sections of the first part, only two changes need to be made (other than replacing the word knot with the word link anywhere this word appears).

First, one needs to change the formal definition (Section An Introduction to Knot Theory), since links may be defined as functions from a union of circles (instead of just one circle) to the 3-sphere.

Second, when discussing The Alexander-Conway Polynomial, P(1) = 1 is not necessarily valid. Indeed, P(1) = 1 only when the link consists of just one component (i.e. is a knot). If the link consists of two or more components, P(1) = 0.

The rest of the discussion of the first part is still valid. Two links are ambient isotopic iff they can be connected through a series of Reidemeister moves, and thus all knot invariants are also link invariants. In addition, the "braid closure" theorem is still valid: all links can be created through the closure of appropriate braids, and two links are equivalent iff the corresponding braids can be connected through the Markov (or the "L") moves. Finally, the fundamental group of the complement of a link (a.k.a. the link group), can be presented through the method introduced by Wirtinger.

When dealing however with the problem of link classification on the other side, almost nothing remains the same, and every one of the seventeen sections needs either to be modified, or completely removed. To continue with his discussion, and to learn more about how the tabulation of links differs from the tabulation of knots, the reader is referred to this page.

Charilaos Aneziris

Copyright 2001


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