Knot theory is a branch of algebraic topology where one studies what is known as the placement problem, or the embedding of one topological space into another. The simplest form of knot theory involves the embedding of the unit circle into three-dimensional space. For the purposes of this document a knot is defined to be a closed piecewise linear curve in three-dimensional Euclidean space R

Two knots or links are considered
equivalent if one can
be smoothly
deformed into the other, or equivalently,
if there exists a
*homeomorphism*
on R^{3} which maps the image
of the first knot onto the second.
Cutting the knot or allowing it to
pass through itself are not permitted.
In general it is very difficult problem to decide if two
given knots are equivalent,
and much of knot theory is
devoted to developing techniques to aid in answering this
question.
Knots that are equivalent to polygonal paths in
three-dimensional space are called *tame.*
All other knots are known as
*wild.*
Most of knot theory concerns only tame knots, and
these are the only knots examined here.
Knots that are equivalent to the unit circle
are considered to be unknotted or trivial.

The simplest non-trivial knot is the trefoil knot which comes in a left and a right handed form.

It is not too difficult to see (but slightly more
difficult to prove) that the trefoil is not
equivalent to the unknot.
Also, the right and left handed versions of the trefoil
are
only equivalent if the
homeomorphism
mapping one into the other
includes a reflection
(other knots, such as the Figure-8 knot
*are* equivalent to their mirror images, these
knots are known as
achiral knots).

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Knots such as the square knot
are usually excluded from knot tables because they
can be constructed of simpler knots.
Knots that cannot be split into two or more simpler
knots are called * prime.*

It is not always easy to tell when a given picture of a knot is the simplest possible picture for that knot. For example the following is the unknot:

The interested reader may enjoy trying to untangle this beast by making a sequence of knot diagrams. If you give up, look at MPEG animation of the untangling (106 kbytes). This unknot image is also available in raycasted and raytraced 608x608 pixel JPEG versions (about 130 kbytes each).

- Trefoil (MPEG, 50 kilobytes)
- Figure 8 knot (MPEG, 57 kilobytes)
- 6.3 knot (MPEG, 95 kilobytes)
- An animation of a six-component Brunnian link falling apart after one component has been removed (380 kilobytes).

- Suspended knots --- A ordinary knot in 3D can be
*suspended*in 4D to create a knotted 2-sphere. - Spun knots --- One problem with suspended knots is that they
are not smooth at the poles.
While this might not seem like an important fact, it is
actually not possible in general to smooth out the poles.
One way to get a smoothly embedded sphere in 4D is to
*spin*a 3D knot about a plane in 4D. - Twist-spun knots --- A generalization due to Zeeman
of spinning. This method produces knot types that
that cannot be produced by ordinary spinning
**Projection of a spun knot in 4d with transparent bands on the surface. Look at an MPEG animation the same knot rotating rigidly in 4d (418 kbytes).**

Some strange things happen in higher dimensions with links, see the Linking Spheres page.

- Untangling the Mathematics of Knots, part of the wonderful MegaMath project.
- This page of mine has links to other people in the business of relaxing or drawing knots.
- An incomplete but growing list I've made of books about knot theory.
- Peter Suber has compiled an excellent page about Knots on the Web, there you'll find links to sites about knots from many different perspectives, from the practical to the purely aesthetic and abstract.
- Another great site with knot-related resources listed is the Ropers Knots Page with an extensive page of links to all sorts of other knot sites on the WWW.

Copyright © 1998-2000 by Robert G. Scharein