Gourt > Science > Math > Topology > Knot Theory (1)
Knot theory is a branch of topology inspired by observations, as the name suggests, of common knots. But progress in the field does not depend exclusively 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.
When mathematical topologists consider knots and other entanglements such as links and braids, they describe how the knot is positioned in the space around it, called the ambient space. If the knot is moved smoothly to a different position in the ambient space, then the knot is considered to be unchanged, and if one knot can be moved smoothly to coincide with another knot, the two knots are called "equivalent".
In mathematical language, knots are embeddings of the circle in three-dimensional space. A mathematical knot thus resembles an ordinary knot with its ends spliced. The topological theory of knots investigates such questions as whether two knots can be smoothly moved to match one another, without opening the splice. The question of untying an ordinary knot has to do with unwedging tangles of rope pulled tight, but this concept plays at best a minor role in the mathematical theory. A knot can be untied in the topological sense if and only if it can be smoothly moved through the ambient space until it assumes the shape of a circle. If this can be done, the knot is called the unknot.
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Knots on the Web (Peter Suber) - The most comprehensive collection of knotting resources on the web. Sections on knot tying, mathematical knot theory, knot art, and knot books.
Meta Description: [ The most comprehensive collection of knotting resources on the web. Sections on knot tying, mathematical knot theory, knot art, knot discussion forums, knot software, knot videos, and knot books. Also a knot gallery. ]
A Circular History of Knot Theory - Starting with the flawed theory of Kelvin's knotted vortex to the work of Thurston, Jones and Witten, knot theory has circled back to its ancestral origins of theoretical physics.
A Knot Theory Primer - Comprehensive knot theory site focusing on the knot classification problem and knot tabulations. Has a tabulation of knots with up to 12 crossings.
A Third Year Lecture Course on Knots - Includes examples, solutions, knot tables, pretty pictures. Course material includes: colouring, Alexander and Jones polynomials, tangles and braids.
404
An Introduction to Knot Theory - Introductory level tutorial requiring only a high school mathematics background, some linear algebra is needed in places.
500
BraidLink - Braidlink is software for knot and braid theory computations. It performs both analytic and numerical manipulations of knots and braids.
Cook's Borromean Ring Links - Links to pages and two outlines of proofs that show the Borromean rings can't be made from circular rings.
Geometry and the Imagination - Has a small section on knot theory at an introductory level. Also has sections on orbifolds, polyhedra and topology.
Meta Description: [ Geometry and the Imagination ]
404
Harmonic Knots - An introduction to harmonic knots. Gives (parametric) formulas for knots of up to 7 crossings.
History of Knot Theory - Biographies of early knot theorists. Many early papers on knot theory (in pdf format) including papers by Tait, Kirkman, Little and Thomson.
Kauffman, Louis H - A topologist working in knot theory discusses the connection between knot theory and statistical mechanics. Sections on cybernetics and knots, Fourier knots and the author's research papers.
Knot Plot - A collection of knots and links, viewed from a (mostly) mathematical perspective. Nearly all of the images here were created with KnotPlot, a program to visualize and manipulate mathematical knots in three and four dimensions.
Meta Description: [ The KnotPlot Site, a visual exploration of mathematical knots. ]
Knot Theory - An overview of knot theory from Mathworld
Knot Theory - Covers techniques of distinguishing knots, types, applications, and Conway notations. Includes illustrations.
404
Knot Theory Group University of Liverpool - Links to preprints and to programs written in pascal for doing knot calculations.
Meta Description: [ Summary ]
Knot Theory Invariants: The HOMFLY Polynomial - A brief article on the HOMFLY polynomial and how it is calculated.
Knot Theory Online - This site is designed for mathematics students at the high school and college levels as an introduction to an area of mathematics seldom explored in the typical math classroom - the Theory of Knots.
Knotscape - By Jim Hoste and Morwen Thistlethwaite. Provides convenient access to tables of knots. Linux, Solaris.
Mathematics and Knots Exhibition - High school level introduction to knot theory. Covers colourings, connected sums, torus knots, prime knots and applications of knot theory.
Megamath Knot Theory Page - An introductory overview of knot theory.
Morwen Thistlethwaite's Home Page - Has many beautiful images of symmetric knots, and information about a computer program called Knotscape (compiled binaries for Linux, Sunos and Alpha platforms). Includes pictures of knots with 13 crossings or less.
New Knot Tables - Covers families of knots of p, pq, p1q, p11q, p111q, pqr, pq1r types. Explains properties and notations. Includes diagram photos.
Pictures of Knots - A table of graphics of all knots of up to nine crossings. Also includes pictures of some links.
String Figure Mathematics (or Trivial Knot Theory) - A mathematical analysis of string figures. Theorems, examples, illustrations and conjectures on patterns created with an unknotted string.
The Geometry Junkyard: Knot Theory - A page of links on geometric questions arising from knot embeddings.
The Knot Theory Home Page - Elementary introduction to knot theory. Covers the existence of knots, Reidemeister moves and colorations.
The KnotPlot Site - Has a large number of beautiful graphics of knots created with KnotPlot. Contains an introductory section on mathematical knot theory. KnotPlot software for various platfroms can be downloaded.
Thomas Fink (Tie Knots) - Thomas Fink and Yong Mao, used ideas from statistical mechanics to show there are 85 ways to tie a tie. They discovered a number of new aesthetically pleasing tie knots. This page has links to their original papers and to their book ``The 85 Ways to Tie a Tie''.
Meta Description: [ Quality. ]
Using Topology to Probe the Hidden Action of Enzymes - Describes how knot theory is used to understand the action of enzymes that affect DNA topolgy (in pdf format).
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