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 principle could be assumed by a loop of string.
In mathematical jargon, knots are embeddings of the closed circle in three-dimensional space.
History
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' and 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] (http://www.ma.hw.ac.uk/RSE/meetings_etc/ordmtgs/2001/reports/knots.htm) [2] (http://www.kpbsd.k12.ak.us/kchs/JimDavis/CalculusWeb/Knot%20Theory%20History.htm) The vortex theory has been disregarded by some, but the general knot theory has grown into a subject with wide and often unexpected applications, for example to theories of particle physics, DNA replication and recombination, and to areas of statistical mechanics.
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.
 |
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
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:
I. Twist and untwist in either direction.
II. Move one loop completely over another.
III. 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.
See also
Further reading
- The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Colin Adams, 2001, ISBN 0716742195
- Knots: Mathematics With a Twist, Alexei Sossinsky, 2002, ISBN 0674009444
- Knot Theory, Vassily Manturov, 2004, ISBN 0415310016
- MathWorld: Reidemeister Moves (http://mathworld.wolfram.com/ReidemeisterMoves.html)
Other resources
- Software for Viewing Knots (Freeware) (http://www.pims.math.ca/knotplot/download.html)
de:Knotentheorie fr:Théorie des nœuds it:Teoria dei nodi ja:結び目理論