Knot theory
Note: This page is part of the
KnotPlot Site,
where you'll find many more pictures of knots and links as well as MPEG animations
and lots of things to download.
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 R3.
Two or more knots together are called a link.
Thus a mathematical knot is somewhat different from the
usual idea of a knot, that is, a piece of string
with free ends.
The knots studied in knot theory are (almost) always
considered to be closed loops.
Two knots or links are considered
equivalent if one can
be smoothly
deformed into the other, or equivalently,
if there exists a
homeomorphism
on R3 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).
Knot tables
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Knots have been
catalogued in order of increasing complexity.
One measure of complexity that is often used is the
crossing number, or the number of double points in the simplest
planar projection of the knot.
There is only one knot with crossing number three
(ignoring mirror reflections),
the trefoil or cloverleaf knot.
The figure-8 knot is the only knot with a crossing number of four.
There are two knots with a crossing number of five, three with a
crossing number of six, and seven knots with a crossing number of
seven.
From there on the numbers increase dramatically.
There are 12,965 knots with 13 or fewer crossings in a minimal
projection and 1,701,935 with 16 or fewer crossings.
Following are pictures of the sixteen simplest knots:
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, watch one of movies below:
Untying the Unknot
Untying the Unknot, anaglyph version (needs red/blue 3D glasses)
This unknot image is also available in
raycasted and
raytraced
608x608 pixel JPEG
versions (about 130 kbytes each).
KnotPlot Relaxations
In the first three of the
following animations, the starting configuration of the
knot is specified by the Conway notation.
The knot then relaxes under the simple
dynamics used by KnotPlot (the arrows
are force vectors).
More information and animations will be available soon.
Braid theory
See the companion page on
braid theory (incomplete).
Higher dimensional knot-theory
Knot theory is extendible into higher dimensions.
Generalized knot theory considers embeddings of the
(N - 2)-dimensional sphere into an N dimensional
sphere.
In 4D knot theory we consider the embeddings of 2-spheres.
There are a number of ways to construct knotted 2-spheres
in 4D.
- 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.
More about knots and links
To learn more about knot theory, go to one of the following sites:
Go to the
KnotPlot Site.
Copyright
©
1998-2000 by
Robert G. Scharein
