Knot theory

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

Knot theory is a branch of topology inspired by observations, as the name suggests, of common knots. Knot theory concerns itself with abstract properties of theoretical knots — the spatial arrangements that in principle could be assumed by a loop of string.

When mathematical topologists consider knots and other entanglements such as links and braids, they describe how the knot is positioned in the space around it, called the ambient space. If the knot is moved smoothly to a different position in the ambient space, then the knot is considered to be unchanged, and if one knot can be moved smoothly to coincide with another knot, the two knots are called "equivalent".

In mathematical language, knots are embeddings of the circle in three-dimensional space. A mathematical knot thus resembles an ordinary knot with its ends spliced. The topological theory of knots investigates such questions as whether two knots can be smoothly moved to match one another, without opening the splice. The question of untying an ordinary knot has to do with unwedging tangles of rope pulled tight, but this concept plays at best a minor role in the mathematical theory. A knot can be untied in the topological sense if and only if it can be smoothly moved through the ambient space until it assumes the shape of a circle. If this can be done, the knot is called the unknot.

Modern knot theory has extended the concept of a knot to higher dimensions. One recent application of knot theory has been to the question of whether two strands of DNA are equivalent without cutting.

A more complicated knot
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A more complicated knot

Contents

[edit] History

Knot theory originated in an idea of Lord Kelvin's (1867), that atoms were knots of swirling vortices in the æther. 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. We now know that this idea was mistaken, and that the discrete wavelengths depend on quantum energy levels.[1]

Scottish physicist Peter Tait spent many years listing unique knots in the belief that he was creating a table of elements. When the luminiferous æther was not detected in the Michelson-Morley experiment, vortex theory became completely obsolete, and knot theory ceased to be of great scientific interest. Following the development of topology in the late nineteenth century, knots once again became a popular field of study. Today, knot theory finds applications in string theory and loop quantum gravity, in the study of DNA replication and recombination, and in areas of statistical mechanics.

[edit] 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. Although a complicated algorithm exists to decide in every case whether two such embeddings are different or the same, a major issue in the theory is to understand how hard this problem really is.[1]

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.

[edit] Knot diagrams

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. When this is the case, we 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.

[edit] Reidemeister moves

Main article: Reidemeister move
The 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. Many important invariants can be defined in this way, including the Jones polynomial. Older examples of knot invariants include the fundamental group and the Alexander polynomial.

[edit] Knots in higher dimensions

In four dimensions, any closed loop of one-dimensional string is equivalent to an unknot. We can achieve the necessary deformation in two steps. The first step is to "push" the loop into a three-dimensional subspace, which is always possible, though technically difficult. The second step is uncrossing, which is technically easy. Suppose one strand is behind another as seen from a chosen point. Lift it into the fourth dimension, so there is no obstacle (the front strand having no component there); then slide it forward, and drop it back, now in front. An analogy for the plane would be lifting a string up off the surface.

In general, piecewise-linear n-spheres form knots only in (n+2)-space (a result of E. C. Zeeman), although one can have smoothly knotted n-spheres in (n+3)-space for n > 2 (independent results of A. Haefliger and Jerome Levine).

[edit] Adding knots

Two knots can be added by cutting both knots and joining the pairs of ends. This can be formally defined as follows: consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots. Form a new knot by deleting the first pair of opposite sides and adjoining the other pair of opposite sides. The resulting knot is the sum of the original knots.

This operation is called the knot sum, or sometimes the connected sum or composition of two knots. Knots in 3-space form a commutative monoid with prime factorization, which allows us to define what is meant by a prime knot. The trefoil knot is the simplest prime knot. Higher dimensional knots can be added by splicing the n-spheres. While you cannot form the unknot in three dimensions by adding two non-trivial knots, you can in higher dimensions, at least when one considers smooth knots in codimension at least 3.

[edit] Tabulating knots

Knot diagrams are useful visual aids, but they are clumsy to work with in terms of establishing equality between different knots. Many notations have been invented for knots, the following are among the more useful and widely used.

[edit] The Dowker notation

A knot diagram with crossings labelled for a Dowker sequence
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A knot diagram with crossings labelled for a Dowker sequence

The Dowker notation (or Dowker sequence) for a knot is sequence of even integers. To generate the Dowker notation, traverse the knot using an arbitrary starting point and direction. Label each of the n crossings with the numbers 1 ... 2n in order of traversal (each crossing is visited and labelled twice), with the following modification: if the label is an even number and the strand followed crosses over at the crossing, then change the sign on the label to be a negative. When finished, each crossing will be labelled a pair of integers, one even and one odd. The Dowker notation is the sequence of even integer labels associated with the labels 1, 3, ... 2n-1 in turn.

For example, a knot diagram may have crossings labelled with the pairs (1,6) (3,-12) (5,2) (7,8) (9,-4) and (11,-10). The Dowker notation for this labelling is the sequence: 6 -12 2 8 -4 -10.

A knot can be recovered from a Dowker sequence, but the recovered knot may differ from the original by being a reflection or (more generally) by having any connected-sum component reflected in the line between its entry/exit points - the Dowker notation is unchanged by these reflections. Knots tabulations typically consider only prime knots and disregard chirality, so this ambiguity does not affect the tabulation.

[edit] Conway's notation

Conway's notation is based on the construction of knots. Say a knot can be constructed in the following way:

  1. begin with two parallel strings, one above the other (such that the strings are horizontal)
  2. give the strings 2 left-handed twists (the bottom string goes over the top string in the first twist)
  3. reflect the strings through the NW to SE diagonal line
  4. give the strings 3 right-handed twists
  5. connect the top two string ends and the bottom two string ends

Then this knot is denoted by the sequence 2 -3 (the numbers indicate the number of left-handed twists). In any given sequence, a NW to SE line reflection is implied between every set of twists.

The usefulness of Conway's notation lies in computing the continued fraction that corresponds to the sequence. If you have two knots whose Conway notation works out to the same continued fraction, then the knots are equivalent.

[edit] Signed planar graphs

This notation links knot theory and graph theory. Once the signed planar graph corresponding to a particular knot is known, questions about knots become questions about graphs. This has applications in commerce and statistical mechanics.

[edit] See also

[edit] References

  • M.Sh. Farber, A.V. Chernavskii, "Knot theory" SpringerLink Encyclopaedia of Mathematics (2001)
  • Colin Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, (2001), ISBN 0-7167-4219-5
  • Dale Rolfsen, Knots and Links, (1976), ISBN 0-914098-16-0

[edit] Resources

[edit] Footnotes

  1. ^ Peterson, Mathematical Tourist, 1988, p74
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