| 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 |
| Knot_Theory Covers techniques of distinguishing knots, types, applications, and Conway notations. Includes illustrations. |
| Knot_Theory An overview of knot theory from Mathworld |
| Knot_Theory_Group_University_of_Liverpool Links to preprints and to programs written in pascal for doing knot calculations. |
| The_Knot_Theory_Home_Page Elementary introduction to knot theory. Covers the existence of knots, Reidemeister moves and colorations. |
| 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 T |
| 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. |
| 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. |
| 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 cr |
| 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. |
| 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. |
| 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 o |
| 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). |
| ACFnewsource_org\'s_Environment_Coverage Feature stories from U.S. nonprofit organization specializing in research, development and production of news stories on substantive issues, including the environment, for commercial radio and televis |
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A Circular History of Knot Theory
In the nineteenth century physicists were speculating about the underlyingprinciples of atoms. In 1867, Lord Kelvin put forward a comprehensive theoryof atoms which, through heuristic reasoning, seemed to explain several ofthe essential qualities of the chemical elements. Kelvin's theory conjecturedthat atoms were knotted tubes of ether. (To a topologist a knot in 3-space is any closed loop having no self-intersections and a link is anycollection of non-intersecting closed loops.) The topological stabilityand the variety of knots were thought to mirror the stability of matterand the variety of chemical elements. 
Kelvin's theory of vortex atoms was taken seriously for about two decades.Maxwell thought that ``it satisfies more of the conditions than any atomhitherto considers''. This theory inspired the celebrated Scottish physicistPeter Tait to undertake an extensive study and tabulation of knots in anattempt to understand when two knots were ``different''. (The later stagesof this study were in collaboration with C. N. Little.) Tait's intuitiveunderstanding of ``different'' and ``same'' is still a useful notion. Twoknots are isotopic if one can be continuously manipulated in 3-space (no self-intersections allowed) until it looks like the other. Theaccompanying diagram shows a portion of Tait's study---an enumeration of knots and links interms of the crossing number of a plane projection. If Kelvin's theory hadbeen the correct foundation for the classification of the chemical elements,then Tait's knot table would have been the basis for a periodic table ofelements. But Kelvin's theory was fundamentally mistaken and physicistslost interest in the Tait's work. 
What the physicists abandoned, intrigued mathematicians, then and now, andthe basic question is still the same: how do we tell when two knots are isotopically the same? (Research tip: Sometimes the most interesting problems can be found insomeone else's trash.) This failed atomic theory also left in its wake theriches of Tait's tabulation---163 knot projections---and a rudimentary understandingof isotopic sameness in terms of how one projection could be continuouslymanipulated to look like another. This understanding of projection manipulationwas summarized in a set of conjectures for knot projections, the famous Tait Conjectures. 
To attack the Tait Conjectures and the basic question of sameness of knots,topologists developed knot invariants. An early example of a successful knot invariant is the Alexander polynomial, discovered by J. W. Alexander in 1927. The Alexander polynomial for theknot labeled 3_1 (the trefoil) is -(txt)+t -1 and the polynomial for 4_1 (the figure-eight) is -(txt)+3t-1. Since these two polynomials are different we know their associated knotsare different. The Alexander polynomial was remarkable for how successfulit was in distinguishing the knots in Tait's orginal table and it gave witnessto how thorough a researcher Tait was. (Historical note: The last of thefew duplications in the Tait/Little table was found in 1974 by Kenneth Perko,a New York lawyer and part-time topologist, while he was manipulating loopsof rope on his living room floor. If a lawyer can do research in knot theory,it can't be that hard.) Unforunately, there are many knots with equivalentAlexander polynomial that can be shown to be isotopically different throughthe uses of other invariants. 
So the search was on for more sensitive knot invariants that would detectwhen two knots were different. This led to alternate understandings of thenotion of sameness. In particular, to a topologist there is no differencebetween the loops representing 4_1 and 5_1. What is different is the space away from these loops, that is the complement of the knot. Two topological spaces are homeomorphic if there is a bijective invertible continuous function that maps one spaceto the other. Thus, we have an alternate notion of sameness: if two knots/linkshave homeomorphic knot/link complements then they are homeomorphic knots/links. Now, it would seem that homeomorphic sameness would be weaker than isotopicsameness. And in fact, for link complements it is---there exist examplesof links that are not isotopic, but have homeomorphic complements. But forknots a seminal result of Cameron Gordon and John Luecke showed that twoknot are homeomorphic if and only if they are isotopic. In the vernacularof the knot theorist, a knot determines its complement. 
Understanding that the principle object of study is the knot complementplaces knot theory inside the larger study of 3-manifolds. A 3-manifold is a space which locally (assume you are near sighted) looks likestandard xyz-space and knot complements are readily seen as examples of 3-manifolds. It was through the study of 3-manifolds that in the 1970's knot theory began returning to its ancestoralroots in physics. To understand this we have to flashback to the 1860'swork of Bernhard Riemann. Riemann was interested in relating geometric structuresto the forces in physics. Building on Gauss' work, Riemann investigatedthree different geometric structures for 3-dimensional spaces---elliptic, euclidean, and hyperbolic. (Einstein's Theoryof Relativity was built on Riemannian geometry.) Each of these distinctstructures can be characterized by the behavior of triangles in planes.In elliptic 3-space, the interior angles of a triangle in a plane have a sum greater than 180 degrees. In Euclidean 3-space, the sum is 180 degrees and in hyperbolic 3-space the sum is less than 180 degrees. In 1978, William Thurston established sufficient conditions forwhen a 3-manifold possesses a hyperbolic structure. Surprisingly, except for a wellunderstood subclass of knots, all knot complements possess a complete hyperbolicstructure. (The beauty of Thurston's work is captured in the video Not Knot that is distributed by the American Mathematical Society and has been frequentlyviewed at Grateful Dead concerts.) 
Thurston's work on hyperbolic structures firmly re-established knot theory'sconnections with physics. In the 1980's, through some totally unexpectedroutes, knot theory made further connections with its ancestral roots. In1987 Vaughan Jones discovered a totally different polynomial invariant fromthat of Alexander using the theory of operator algebras. Within a shortperiod of time, more than five new polynomial invariants generalizing theJones polynomial were discovered. (One of these polynomial was simultaneouslydiscovered by six different mathematicians and its name is an acronym oftheir last names---HOMFLY.) Moreover, Jones' polynomial quickly led to theproofs that established all of Tait's original conjectures on knot projections. 
With this poliferation of new polynomials it was natural to ask whetherany of these invariants had a natural extensions to all 3-manifolds. Two facts worked in favor of having such extensions: 1) all 3-manifolds can be describe in terms of knots and links via an operation called Dehn surgery; 2) there exists a set of moves, the Kirby calculus, that allow one tomove between differing Dehn surgery descriptions of the same homeomorphic 3-manifold. Using the Kirby calculus as a means to generalizing the polynomialinvariants, Edward Witten, a theoretical physicist, proposed new invariantsfor 3-manifolds. His invariants came out of the theoretical area of physics knowas quantum field theory. These new invariants can be realized as certainaverages of link polynomials obtained from a given Dehn surgery representationof the manifold. 
Starting with the flawed theory of Kelvin's knotted vortex to the work ofThurston, Jones and Witten, knot theory has circled back to its ancestralorgins of theoretical physics. Note; If you are interested in reading more about Knot Theory and 3-manifolds, Dale Rolfsen's book, Knots and Links, is a good introductory source.WWM, 4/1/1999