Knot theory

Missing image
Trefoil knot, the simplest non-trivial knot.

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 principle could be assumed by a loop of string.

In mathematical jargon, knots are embeddings of the closed circle in three-dimensional space. An ordinary knot is converted to a mathematical knot by splicing its ends together. The topological theory of knots asks whether two such knots can be rearranged to match, without opening the splice. The question of untying an ordinary knot has to do with unwedging tangles of rope pulled tight. A knot can be untied in the topological theory of knots if and only if it is equivalent to the unknot, a circle in 3-space.



Knot theory originated in an idea of Lord Kelvin's (1867), that atoms were knots of swirling vortices in the æther (also known as 'ether'). He believed that an understanding and classification of all possible knots would explain why atoms absorb and emit light at only the discrete wavelengths that they do (i.e. explain what we now understand to depend on quantum energy levels). [1] ( Scottish physicist Peter Tait spent many years listing unique knots under the belief that he was creating a table of elements. When ether was discredited through the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory fell out of scientific interest. Only in the past 100 years, with the rise of topology, have knots become a popular field of study. Today, knot theory is inextricably linked to particle physics, DNA replication and recombination, and to areas of statistical mechanics.

An introduction to knot theory

Creating a knot is easy. Begin with a one-dimensional line segment, 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.

Two unknots
The unknot, and a knot
equivalent to it

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.

Reidemeister moves

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The Reidemeister moves

In 1927, working with this diagrammatic form of knots, J.W. Alexander and G.B. Briggs, and independently Kurt Reidemeister, demonstrated that two knot diagrams belonging to the same knot can be related by a sequence of three kinds of moves on the diagram, shown right. These operations, now called the Reidemeister moves, are:

  1. Twist and untwist in either direction.
  2. Move one loop completely over another.
  3. Move a string completely over or under a crossing.

Knot invariants can be defined by demonstrating a property of a knot diagram which is not changed when we apply any of the Reidemeister moves. Some very important invariants can be defined in this way, including the Jones polynomial.

Higher dimensions

You can unknot any circle in four dimensions. There are two steps to this. First, "push" the circle into a 3-dimensional subspace. This is the hard, technical part which we will skip. Now imagine temperature to be a fourth dimension to the 3-dimensional space. Then you could make one section of a line cross through the other by simply warming it with your fingers.

In general piecewise-linear n-spheres form knots only in n+2 space, although one can have smoothly knotted n-spheres in n+3 space.

Adding knots

Two knots can be added by breaking the circles and connecting the pairs of ends. Knots in 3-space form a commutative monoid with prime factorization. The trefoil knots are the simplest prime knots. Higher dimensional knots can be added by splicing the spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions.

See also

Further reading

Other resources

fr:Théorie des nœuds it:Teoria dei nodi ja:結び目理論


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