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Knot theory

Knot theory is a branch of topology that was inspired by observations, as the name suggests, of knots. But progress in the field no longer depends on experiments with twine. Knot theory concerns itself with abstract properties of theoretical knots--the spatial arrangements that in priniciple could be assumed by a loop of string. In mathematical jargon, these are embeddings of the closed circle in three dimensional space.

Knot theory has grown into a subject with wide and often unexpected applications, for example to theories of quantum gravity, DNA replication and recombination, and to areas of statistical mechanics.

Table of contents

An introduction to knot theory

Given a one dimensional line, wrap it around itself arbitrarily, and then fuse its two free ends together to form a closed loop. One of the biggest unresolved problems in knot theory is to describe the different ways in which this may be done, or conversely to decide whether two such embeddings are different or the same.

Before we can do this, we must decide what it means for embeddings to be "the same". We consider two embeddings of a loop to be the same if we can get from one to the other by a series of slides and distortions of the string which do not tear it, and do not pass one segment of string through another. If no such sequence of moves exists, the embeddings are different knots.

A useful way to visualise knots and the allowed moves on them is to project the knot onto a plane - think of the knot casting a shadow on the wall. Now we can draw and manipulate pictures, instead of having to think in 3D. However, there is one more thing we must do - at each crossing we must indicate which section is "over" and which is "under". This is to prevent us from pushing one piece of string through another, which is against the rules. To avoid ambiguity, we must avoid having three arcs cross at the same crossing and also having two arcs meet without actually crossing (we would say that the knot is in general position[?] with respect to the plane). Fortunately a small perturbation in either the original knot or the position of the plane is all that is needed to ensure this.
{Note: This will be a lot easier to follow when there are some diagrams here!}

Reidemeister moves

In 1927, working with this diagrammatic form of knots, Kurt Reidemeister[?] demonstrated that all the allowable moves on a knot could be reduced to three kinds of move on the diagram, shown left. These operations, now called the Reidemeister moves, are:

I. Twist and untwist in either direction.
II. Move one loop completely over another.
III. Move a string completely over or under a crossing.

Reidemeister was the first to demonstrate that knots really exist - that is, that there really are knots that are not equivalent to the unknot. He did this by inventing the first knot invariant, demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves.

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