1. 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. http://www.math.buffalo.edu/~menasco/Knottheory.html

A Circular History of Knot Theory In the nineteenth century physicists were speculating about the underlying principles of atoms. In 1867, Lord Kelvin put forward a comprehensive theory of atoms which, through heuristic reasoning, seemed to explain several of the essential qualities of the chemical elements. Kelvin's theory conjectured that atoms were knotted tubes of ether. (To a topologist a knot in space is any closed loop having no self-intersections and a link is any collection of non-intersecting closed loops.) The topological stability and the variety of knots were thought to mirror the stability of matter and 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 atom hitherto considers''. This theory inspired the celebrated Scottish physicist Peter Tait to undertake an extensive study and tabulation of knots in an attempt to understand when two knots were ``different''. (The later stages of this study were in collaboration with C. N. Little.) Tait's intuitive understanding of ``different'' and ``same'' is still a useful notion. Two knots are isotopic if one can be continuously manipulated in space (no self-intersections allowed) until it looks like the other. The accompanying

2. Ideas, Concepts And Definitions An introductory overview of knot theory. http://www.c3.lanl.gov/mega-math/gloss/knots/knots.html

Knot Theory Knot Theory is the mathematical study of knots. A mathematical knot has no loose or dangling ends; the ends are joined to form a single twisted loop. The central problem of knot theory is distinguishing between various knots and classifying them. The best way to learn about knots is to make some knots and play with them. Some important ideas in Knot Theory classifying knots knots as closed circular braids crossings keeping track of crossings ... seifert surfaces See also . . . Knot Coloring Puzzles A Table of Knots to Study and Make Untangling the Mathematics of Knots Knots, Links and Other Mathematical Tangles ... Knot Theory and the NCTM Standards

3. Knot Theory Online - The Web Site For Learning More About Mathematical Knot Theo 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 http://www.freelearning.com/knots/

Welcome to KT (Knot Theory) Online! Here you can learn about a different kind of Mathematics! Links on this site: [HOME] [HISTORY] [INTRO] [ADVANCED] ... KT HOME Main Page KT HISTORY History of Knot Theory INTRO TO KNOTS What are knots? ADVANCED KT Knot Theory in the Real World KT ACTIVITIES Online activities with knots for you to try KNOT FUNNY Interesting facts, knot-knot jokes, and knotty pictures... Welcome to KT (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. One thing that makes Knot Theory so interesting for mathematicians today is the fact that it's such a "new" topic - Knot Theory is a relatively young field with many opportunities for discovery and exploration by mathematicians young and old. Click below to jump to some of the topics you can learn about at KT Online: A Brief History of Knot Theory - From the mathematical surge of interest in knots a little over a century ago to the recent and exciting application of Knot Theory to DNA and synthetic chemistry, you can get an overview of why knots are such a fascination for scientists and mathematicians alike.

4. Liverpool Knot Theory Group: Knot Programs Four programs, each calculating some invariant from a closed braid presentation. Free to download. Pascal. http://www.liv.ac.uk/~su14/knotprogs.html

Liverpool University Knot Theory Group Programs and Procedures Copies of the following programs and procedures can be downloaded for compilation on your own machine. Some basic descriptions and instructions are attached. Queries can be sent by e-mail to morton@liv.ac.uk This list can be found at URL http://www.liv.ac.uk/~su14/knotprogs.html or by following links from http://www.liv.ac.uk/maths/ See also the list of publications of the Liverpool Knot Theory group , where PostScript copies of some recent preprints can be found. There are at present four programs, each calculating some invariant from a closed braid presentation. In each case the major constraint is the number of braid strings; the growth of calculation time with the number of crossings is roughly quadratic, and so typically it is not difficult to deal with braids having about 100 crossings. Three of the programs are in Pascal. In each case the input required is the braid in terms of standard generators, with no separators, for example 123-2-231. The three programs are br9z.p

5. The Survival, Origin And Mathematics Of String Figures Trivial knot theory, history, and a few new designs by the author, Martin Probert. http://website.lineone.net/~m.p/sf/menu.html

The Survival, Origin and Mathematics of String Figures "Museums and other institutions with string figure artefacts", "The British Museum A. C. Haddon String Figures", "The Origin of String Figures", "String Figures and Knot Theory", "21st-Century String Figures", "The Creation of New String Figures", "The String Lady". by Martin Probert "the World's Most Widespread Game" (James Hornell, Discovery, NOTE: The text size on this website can be changed by the user. Please use your browser to adjust. 1) Museums and other institutions with string figure artefacts Begun 1999. Last revised April 2003 Museums and artefacts . An inventory of over 1200 string figure artefacts in more than 20 museums worldwide. The inventory includes string figures mounted on card, string figures on film, string figure photographs, and recordings of string figure songs. Holding institutions include the British Museum, the Harvard University Peabody Museum, the Australian Museum, and many others. "A wonderful project" Museum Archivist, North America (pers. comm.)

6. Knot Theory -- From MathWorld An overview of knot theory from Mathworld. http://mathworld.wolfram.com/topics/KnotTheory.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Topology Knot Theory Braids General Knot Theory Knot Invariants Knot Operations ... Links

7. Knot Theory Elementary introduction to knot theory. Covers the existence of knots, Reidemeister moves and colorations. http://library.thinkquest.org/12295/

8. Knot-Theory.com - The Art And Science Of String Toys Searchable trick database with types, inventors, and descriptions of each trick. http://www.knot-theory.com

May 28, 2005 Engine engine number 9, on the New York Transit Line 6 months since I said I was going to restart the site, and now I'm maybe almost to the point about thinking about getting things started again. Trying to run 5 sites at once is tough. Anyways, this is just a check up to say hi to everyone still visiting the site, and make sure everyone knows Doctor Popular's new site is up and running . Let him provide you with entertainment while I get my crap together over here. KT will rise up and kick ass soon. Mark my words. Posted by qDot at 12:03 AM syndication links Doctor Popular Pink Raiders

9. 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. http://www.earlham.edu/~peters/knotlink.htm

Knots on the Web Artwork credits Unfortunately I'm much too busy with real, paying work nowadays to update Knots on the Web as often as I used to. Consequently, it has far too many dead links and omits far too many good, new sites. I wish I could say that this will change soon, but it won't. Until it does, please use what is still useful here and forgive the weak spots. If you have a specific knot need or question, and the surviving links here don't help you with it, then try posting your question to rec.crafts.knots or running a search on Google Welcome to my collection of knotting resources. My major sections are on Knot Tying Knot Theory , and Knot Art . But knot lovers will understand that these distinctions are artificial. For example, a good practical knot is both a nugget of hard-won technology and a thing of beauty. Decorative knotting can be useful, and in any case requires uncommon dexterity and practical tying ability. Software developed to help mathematical knot theorists has produced some of the most beautiful knot images ever seen. So look at all three sections even if you think your interests are narrow. You might become happily entangled. My fourth section is on Knot Discussion . Use these discussion forums to find answers to your knotting questions and to help others who know less than you do. My fifth section is on Knot Software . You'll be surprised at how knotting software can make it easier for you to learn to tie knots, to explore the mathematical properties of knots, and to create stunning images of knots, including knots never seen on Earth.

10. Contents Comprehensive knot theory site focusing on the knot classification problem and knot tabulations. Has a tabulation of knots with up to 12 crossings. http://www.inst.bnl.gov/~wei/contents.html

Table of Contents A Knot Theory Primer An Introduction to Knot Theory Showing Knot Equivalence Showing Knot Inequivalence ... Proving that the links cannot be decomposed This is the latest addition to this online tutorial. Its goal is to extend the discussion from the study of knots to the study of links You may be interested in my published book, The Mystery of Knots Charilaos Aneziris charilaos_aneziris@standardandpoors.com Educational institutions are encouraged to reproduce and distribute these materials for educational use free of charge as long as credit and notification are provided. For any other purpose except educational, such as commercial etc, use of these materials is prohibited without prior written permission.

11. An Introduction To Knot Theory Introductory level tutorial requiring only a high school mathematics background, some linear algebra is needed in places. http://www.yucc.yorku.ca/~mouse/knots/knots.html

12. The Geometry Junkyard: Knot Theory A page of links on geometric questions arising from knot embeddings. http://www.ics.uci.edu/~eppstein/junkyard/knot.html

Knot Theory There is of course an enormous body of work on knot invariants, the 3-manifold topology of knot complements , connections between knot theory and statistical mechanics, etc. I am instead interested here primarily in geometric questions arising from knot embeddings. Animated 3d curves and knots , Jos Leys. Atlas of oriented knots and links , Corinne Cerf extends previous lists of all small knots and links, to allow each component of the link to be marked by an orientation. Borromean rings don't exist . Geoff Mess relates a proof that the Borromean ring configuration (in which three loops are tangled together but no pair is linked) can not be formed out of circles. Dan Asimov discusses some related higher dimensional questions. Matthew Cook conjectures the converse Are Borromean links so rare? S. Javan relates the history of the links and describes various generalizations with more than three rings. For more history and symbolism of the Borromean rings, see Peter Cromwell's web site . See also Paul Bourke's Borromean rings page Borromean paper clips A Brunnian link . Cutting any one of five links allows the remaining four to be disconnected from each other, so this is in some sense a generalization of the Borromean rings. However since each pair of links crosses four times, it can't be drawn with circles.

13. Liverpool Pure Maths: Knot Theory Links to preprints and to programs written in pascal for doing knot calculations. http://www.liv.ac.uk/PureMaths/MIN_SET/CONTENT/RESEARCH_GROUPS/knots.html

Pure Mathematics Knot Theory Dept. Home Page Find someone? Pure home page Knot Theory at Liverpool Members of the Research Group Hugh Morton (e-mail: morton@liv.ac.uk Peter Cromwell (e-mail: spmr02@liv.ac.uk Victor Goryunov (e-mail: goryunov@liv.ac.uk Elisabetta Beltrami (e-mail: elisa@liv.ac.uk or beltrami@gauss.dm.unipi.it Pedro Manchon (e-mail: pmanchon@liv.ac.uk Research Students Sascha Lukac (e-mail: lukac@liv.ac.uk Richard Hadji (e-mail: rhadji@liv.ac.uk Previous Members Anna Aiston Yongju Bae (e-mail: ybae@knu.ac.kr Jonny Hill Ian Nutt (e-mail: iann@wrox.com Marta Rampichini (e-mail: rampichini@mat.unimi.it Hayley Ryder Paul Strickland (e-mail: p.m.strickland@livjm.ac.uk Publications We maintain a list of our publications , which includes PostScript copies of some recent preprints. There are also some braid programs , mainly in Pascal, for calculating a number of knot invariants. Areas of Interest Hugh's interests include: Fibred knots and links.

14. 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. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. Knot Theory Online - The Web Site For Learning More About Mathematical Knot Theo There are many great knot theory sites on the Internet. You can jump directly to the knot theory page for tons of information on the subject. http://www.freelearning.com/knots/fun.htm

Knot Funny Have a little bit of "knotty" fun along your journey to becoming a better mathematician. Links on this site: [HOME] [HISTORY] [INTRO] [ADVANCED] ... KT HOME Main Page KT HISTORY History of Knot Theory INTRO TO KNOTS What are knots? ADVANCED KT Knot Theory in the Real World KT ACTIVITIES Online activities with knots for you to try KNOT FUNNY Interesting facts, knot-knot jokes, and knotty pictures... "Knotty" Fun for All This page is packed with fun and interesting diversions for you to enjoy as you learn more about the world of Knot Theory: 1) "Cut that out!" - Fun activity with a Moebius strip that becomes one of our knot friends. 2) The Dancing Knot - Test your knot-making skill with this activity. 3) Tangled Hands - Can you make a knot without letting go of the ends of the string? 4) Human Knots - A fun group activity to try with your friends. 5) Knot-knot Joke - A little clean knot humor. "Three strings walk into a bar..." 6) "Links" to Other Great Knot Sites

16. Megamath Knot Theory Page An introductory overview of knot theory. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

17. Mathematical Knots The knots studied in knot theory are (almost) always considered to be closed loops. Most of knot theory concerns only tame knots, and these are the only http://www.cs.ubc.ca/nest/imager/contributions/scharein/knot-theory/knot-theory.

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 R . 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 R 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.

18. Links To Low-dimensional Topology Topics General, Conferences, Pages of Links, knot theory, 3manifolds, Journals. http://www.math.unl.edu/~mbritten/ldt/ldt.html

Links to low-dimensional topology Any comments/suggestions? Send them to me! Enter your comments here: Most recent additions: hard to say, I've stopped keeping track... This page was getting just a little too large, so I've cut it into pieces. General Conferences Pages of Links Knot Theory ... Home pages

19. Untangling The Mathematics Of Knots One of the most peculiar things which emerges as you study knots is how a category of objects as simple as a knot could be so rich in profound http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

20. [hep-th/9905057] Chern-Simons Gauge Theory: Ten Years After A brief review on the progress made in the study of ChernSimons gauge theory since its relation to knot theory was discovered ten years ago is presented. http://arxiv.org/abs/hep-th/9905057

High Energy Physics - Theory, abstract hep-th/9905057 From: Jose M. F. Labastida [ view email ] Date: Sat, 8 May 1999 09:52:13 GMT (626kb) Chern-Simons Gauge Theory: Ten Years After Authors: J. M. F. Labastida Comments: 62 pages, 21 figures, lecture delivered at the workshop "Trends in Theoretical Physics II", Buenos Aires, November 1998 Subj-class: High Energy Physics - Theory; Geometric Topology Full-text: PostScript PDF , or Other formats References and citations for this submission: SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted); CiteBase (autonomous citation navigation and analysis) 1 trackback What's this? (send trackbacks to http://arxiv.org/trackback/hep-th/9905057) Which authors of this paper are endorsers? Links to: arXiv hep-th find abs

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