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Knot theory is a branch of the mathematical field of topology, concerning the study of mathematical knots, i.e. an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations (isotopies). This is basically equivalent to a conventional knot with the ends of the string joined together to prevent it from becoming undone.
Knots can be described in a variety of ways, often through planar diagrams. Given a method of description, a knot will have many descriptions, e.g., many diagrams, representing it. An important way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot. Two knots can be shown to be different if an invariant takes different values on them; however, an invariant may take the same value on different knots.
The concept of a knot has been extended to higher dimensions by considering n-dimensional spheres in m-dimensional Euclidean space. This was investigated most actively in the period 1960-1980, when a number of breakthroughs were made. In recent years, low dimensional phenomena has garnered the most interest.
Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, initiated by Max Dehn, J.W. Alexander, among others, concerns primarily knot invariants such as the knot group or those coming from homology theory, such as the Alexander polynomial.
Modern knot theory has developed deep connections to mathematical methods in subjects such as statistical mechanics and quantum field theory. This began with Jones' discovery of the Jones polynomial in 1984 and continued with contributions from Witten, Kontsevich, and others. A plethora of knot invariants have been invented since, including the quantum invariants, finite type invariants. These have been shown to be connected to more general invariants of 3-manifolds.
In the last 30 years, knot theory has also become a tool in applied mathematics. Biologists use knot theory to understand whether two strands of DNA are equivalent without cutting and to understand the actions of enzymes on DNA. Other knot theoretic techniques have been used to investigate the chirality of molecules.
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[edit] History
Knots were studied by Carl Friedrich Gauss, who imparted his interest to his student Johann Benedict Listing, who furthered their study. The early, significant stimulus in knots would be later, however, by Lord Kelvin's theory of vortex atoms. The theory stated that atoms were knots of swirling vortices in the æther.
He came to this belief in 1867 after observing Scottish physicist Peter Tait's experiments involving smoke rings (see a modern recreation of this experiment. Tait had been inspired by a paper of Helmholtz's on vortex-rings in incompressible fluids.[1] Thomson and Tait 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. Modern physics demonstrates this idea was mistaken, and that the discrete wavelengths depend on quantum energy levels.[1]
Tait subsequently spent many years listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. The conjectures spurred research in knot theory, and were only finally resolved in the 1990s. 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 early 20th century, topologists such as Max Dehn, J.W. Alexander, and Kurt Reidemeister, investigated knots. Out of this sprang the Reidemeister moves, the Alexander polynomial. Dehn also developed Dehn surgery, which related knots to the general theory of 3-manifolds and formulated the word problem in group theory. Early pioneers in the first half of the 20th century include Ralph Fox, who popularized the subject. In this early period, knot theory primarily consisted of study of the knot group, or homological invariants of the knot complement.
Interest in knot theory grew significantly after Vaughan Jones' discovery of the Jones polynomial. This led to other knot polynomial such as the bracket polynomial, HOMFLY polynomial, and Kauffman polynomial and led to Jones-Witten invariants and quantum invariants. Jones was awarded the Fields medal in 1990 for this work.
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] Knot equivalence
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. Perturbing the loop very slightly creates a different embedding of the circle, but intuitively, this embedding should really be considered the "same" as the first. The idea of knot equivalence is to make precise when two embeddings should be considered the same.
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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, without cutting or passing a segment through another, to a different position in the ambient space, then the resulting knot is considered to be equivalent to the original. If one knot can be moved smoothly, as described, to coincide with another knot, the two knots are considered equivalent.
The basic problem of knot theory, the recognition problem, can thus be stated as: given two knots, determine whether or not they are equivalent or not. Algorithms exist to solve this problem, with the first given by Wolfgang Haken. Nonetheless, these algorithms use significantly many steps, and a major issue in the theory is to understand how hard this problem really is.[2] The special case of recognizing the unknot, called the unknotting problem, is of particular interest.
[edit] Knot diagrams
A useful way to visualise and manipulate knots is to project the knot onto a plane - think of the knot casting a shadow on the wall. A small perturbation in the choice of projection place will ensure that the projection is one to one except at the double points, called crossings, where the "shadow" of the knot crosses itself transversely. At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the understrand.
[edit] 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:
- Twist and untwist in either direction.
- Move one loop completely over another.
- Move a string completely over or under a crossing.
[edit] Knot invariants
A knot invariant is a "quantity" that is the same for equivalent knots. In particular, an invariant 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. An elementary example is tricolorability.
Older examples of knot invariants include the knot group, which is the fundamental group of the knot complement, and the Alexander polynomial, which can be computed from the Alexander invariant, a module constructed from the infinite cyclic cover of the knot complement.
An invariant may take the same value on two different knots, so by itself may be incapable of distinguishing all knots. An open problem is what constitutes a complete set of invariants, i.e. completely distinguishes a knot from all others. To be useful, these invariants should be computable from a given description of a knot. Note that this is a different problem from the recognition problem mentioned above. Here the concern is on finding a specific kind of algorithm using invariants.
[edit] 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
- (idea here is just to summarize these things, each should really have its own article)
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
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 cheirality, 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:
- begin with two parallel strings, one above the other (such that the strings are horizontal)
- give the strings 2 left-handed twists (the bottom string goes over the top string in the first twist)
- reflect the strings through the NW to SE diagonal line
- give the strings 3 right-handed twists
- 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
- List of knot theory topics
- Braid theory
- Khipu
- Knot invariant
- Knot polynomial
- Legendrian knots
- DNA topology
- Knotane
- Topoisomerase
[edit] References
- The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Colin Adams, 2001, ISBN 0-7167-4219-5
- Knots and Links, Dale Rolfsen, 1976, ISBN 0-914098-16-0
[edit] Further reading
- Knots: Mathematics With a Twist, Alexei Sossinsky, 2002, ISBN 0-674-00944-4
- Knot Theory, Vassily Manturov, 2004, ISBN 0-415-31001-6
- Introduction to Knot Theory, Richard H. Crowell and Ralph Fox, 1977, ISBN 0-387-90272-4
- On Knots, Louis H. Kauffman, 1987, ISBN 0-691-08435-1
- Knot Theory, Charles Livingston, 1996, ISBN 0-88385-027-3
[edit] External links
[edit] History
- Thomson, Sir William (Lord Kelvin), On Vertex Atoms, Proceedings of the Royal Society of Edinburgh, Vol. VI, 1867, pp. 94-105.
- Silliman, Robert H., William Thomson: Smoke Rings and Nineteenth-Century Atomism, Isis, Vol. 54, No. 4. (Dec., 1963), pp. 461-474. JSTOR link
[edit] Knot tables and software
- KnotInfo: Table of Knot Invariants and Knot Theory Resources
- The wiki Knot Atlas - detailed info on individual knots in knot tables
- KnotPlot - software to investigate geometric properties of knots
[edit] Footnotes
- ^ Peterson, Mathematical Tourist, 1988, p74
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