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- Knot Theory

  [Fred Curtis - Mar 2001]
This page is a tiny introduction to Knot Theory. It describes some basic concepts and provides links to my work and other Knot Theory resources.

[What is Knot Theory?] [My Interests] [Old papers I'm typing up] [References]

What is Knot Theory?

Knot theory is a branch of mathematics dealing with tangled loops. When there's just one loop, it's called a knot. When there's more than one loop, it's called a link and the individual loops are called components of the link. A picture of a knot is called a knot diagram or knot projection. A place where parts of the loop cross over is called a crossing. The simplest knot is the unknot or trivial knot, which can be represented by a loop with no crossings.

The big problem in knot theory is finding out whether two knots are the same or different. Two knots are regarded as being the same if they can be moved about in space, without cutting, to look exactly like each other. Such a movement is called an ambient isotopy - the ambient refers to moving the knot through 3-dimensional space, and isotopy is a scary word from topology for the continuous deformation of an object without cutting it or letting it pass through itself.

Kurt Reidemeister showed in 1932 that any diagram of a knot can be turned into any other diagram of the same knot using a kit of 3 moves called the Reidemeister Moves. Unfortunately, for the purposes of working out whether two knots are the same, it can take an enormous number of Reidemeister moves to get from one diagram to another. Any attempt to untangle the Monster diagram with Reidemeister moves involves a temporary increase in the number of crossings.

A special case of working out whether two knots are the same is the problem of working out whether a given knot is the unknot. Wolfgang Haken devised a procedure in 1954, published in 1961, but it takes an enormous amount of effort for any but the simplest inputs [Hass99].

An excellent introductory book for knot theory is [Adams94]. A good on-line introduction is ThinkQuest Knot Theory. A comprehensive page of high-quality sites is Peter Suber's Knot Links.

My Interests

Old papers I'm typing up


  • [Adams94] Colin C. Adams. The Knot Book - An Elementary Introduction to the Mathematical Theory of Knots. W. H. Freeman and Company, New York, 1994.
  • [Hass99] Joel Hass, Jeffrey C. Lagarias, Nicholas Pippenger. The Computational Complexity of Knot and Link Problems. Journal of the ACM, Vol. 46, No. 2 (March 1999), p.185 [Postscript preprint]
A link with two components
The Unknot, the simplest knot, can be drawn with no crossings
The Monster - a diagram of the unknot. Can you untangle it? Give up?
Trefoil knots - these can't be untangled into the unknot, and the two mirror-image versions can't be deformed into one another
Type I Reidemeister Move
Type II Reidemeister Move
Type III Reidemeister Move

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