Knot theory
Knot theory is a branch of mathematics which was originally inspired by studies of physical knots tied in ropes. It is the study of the possible configurations of knots, represented as a continuous loop -- formally, these are embeddings of the closed circle in three dimensional space.
It has grown into an abstract subject in its own right, with unexpected connections to such topics as statistical mechanics and quantum gravity being recently discovered.
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
| The 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.
- See also
- Intro to Knot invariants
- Braid theory
- Still to come:
- Signed Graph representation of knots
- History of knot theory, including resurgence since Jones polynomial
- Maybe try to explain Witten's connection between knots and quantum gravity!