1. Knot Theory Online - The Web Site For Learning More About Mathematical Knot Theo knot theory. knot theory is the mathematical study of knots. 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.

2. Contents An Introduction to knot theory. Showing Knot Equivalence. Showing Knot Inequivalence 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.

3. Mathematical Knots knot theory is a branch of algebraic topology where one studies what is known as the placement problem, or the embedding 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.

4. The KnotPlot Site knot theory. There is of course an enormous body of work on knot invariants, the 3manifold topology of knot complements, http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html

The KnotPlot Site Welcome to the KnotPlot Site! Here you will find a collection of knots and links, viewed from a (mostly) mathematical perspective. Nearly all of the images here were created with KnotPlot, a fairly elaborate program to visualize and manipulate mathematical knots in three and four dimensions. You can download KnotPlot and try it on your computer (see the link below), but first you may want to look at some of the images in the picture gallery. Knot Pictures Check out the mathematical knots M ) page as well to see more knot pictures. Or try some of the following examples to see some knots in a different light. The pages marked with have been updated or created as of 11 Feb 2003. Those marked with an M have at least one MPEG animation. Various Collections The KnotServer (experimental, still not too useful) Knots as radiating tubes M More knots as radiating tubes Fancy Knots and Links, together with Antoine's Necklace (a fractal link) Nifty Knots with Fancy Lighting A knot zoo or two , and assortments of torus knots with neat colours knot chains , and knotted graphs More torus knots and links , arranged by crossing number Brunnian Links M Tangle Calculator Examples Linking Spheres in three or more dimensions PDF and PostScript examples Knot Carpets Celtic Knots 3D Stereoscopic Pictures ( page 1 page 2 page 3 page 4 ... Hopf Torus M Knots on the Cubic Lattice Hyperbolic knots (3D models added 25 Feb 2003) Symmetric projections of minimal-stick knots Simple interactive knot viewer (randomly chosen knot, requires Java)

5. Ideas, Concepts And Definitions An introductory overview of knot theory.Category Science Math Topology knot theory......knot theory. knot theory is the mathematical study of knots. A mathematicalknot play with them. Some important ideas in 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

6. New Ideas About KnotsDiscover Links That Describe Basic Knot Theory. Includes Re The knot theory MA3F2 page Prerequisites Little more than linear algebra plus an ability to visualise objects in 3dimensions. Some knowledge of groups given by generators and relations, and some basic topology would be helpful. http://www.cs.uidaho.edu/~casey931/new_knot

New Ideas about Knots These ideas aren't new to the world, they are new to me (Nancy). They are things that I have discovered since I wrote the basic MegaMath information about knots Recently I've discovered two books about Knot Theory that I can read- without feeling like I had to learn a foreign language and a foreign alphabet. They make me feel perfectly capable of understanding this branch of mathematics. So I have been learning a lot of new things about knot theory. Here are some of my discoveries... Tangled up pictures of tangled up ropes Knot coloring puzzles Knots and counting Pretzel knots

7. The Geometry Junkyard: Knot Theory A page of links on geometric questions arising from knot embeddings.Category Science Math Topology knot theory......The Geometry Junkyard. knot theory. Knots on the Web, P. Suber. Includessections on knot tying and knot art as well as knot theory. 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. 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 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. Colinear points on knots Greg Kuperberg shows that a non-trivial knot or link in R necessarily has four colinear points.

8. Knots Covers techniques of distinguishing knots, types, applications, and Conway notations. Includes illustra Category Science Math Topology knot theory......knot theory. by Rohit Chaudhary. knot theory is a mathematical studyof knots. It is an area of mathematics known as topology. http://www.mapleapps.com/categories/mathematics/Knot theory/html/Knots.htm

9. Liverpool Pure Maths: Knot Theory Links to preprints and to programs written in pascal for doing knot calculations.Category Science Math Topology knot theory......Pure Mathematics knot theory. knot theory at Liverpool. Members ofthe Research Group. Hugh Morton (email morton@liv.ac.uk) Peter 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.

10. Liverpool Knot Theory Group: Publication List Articles and preprints from 1987 onward. Some are available for download in postscript format.Category Science Math Topology knot theory Research Oriented......Liverpool University knot theory Group Articles and Preprints (1987 onwards). J.knot theory Ramif. 1 (1992), 203206. Gzipped PostScript version (42K). http://www.liv.ac.uk/~su14/knotprints.html

Liverpool University Knot Theory Group Articles and Preprints (1987 onwards) Copies of the following papers can be requested by sending e-mail to morton@liv.ac.uk You can download PostScript versions of some of the recent preprints. This list can be found at URL http://www.liv.ac.uk/~su14/knotprints.html or by following links from http://www.liv.ac.uk/maths/ Programs for some knot theory calculations can be found in the list of programs of the Liverpool Knot Theory group You may also enjoy a collection of material on the Borromean Rings H.R.Morton and H.B.Short, `The 2-variable polynomial of cable knots'. Math. Proc. Camb. Philos. Soc. H.R.Morton, `Polynomials from braids'. In `Braids', ed. Joan S. Birman and Anatoly Libgober, Contemporary Mathematics 78, Amer. Math. Soc. (1988), 375-385. H.R.Morton and P.Traczyk, `The Jones polynomial of satellite links around mutants'. In `Braids', ed. Joan S. Birman and Anatoly Libgober, Contemporary Mathematics 78, Amer. Math. Soc. (1988), 587-592. P.R.Cromwell

11. Mouse's Knot Theory Home Page Introductory level tutorial requiring only a high school mathematics background, some linear algebra Category Science Math Topology knot theory......An elementary knot theory tutorial with bibliography and links toother resources. knot theory. This page is part of a project for http://yucc.yorku.ca/~mouse/knots/knots.html

12. Applications Of Knot Theory Applications of knot theory. The applications of knots themselves are familiar,but what good is knot theory? Back to Mouse's knot theory Home Page. http://yucc.yorku.ca/~mouse/knots/applications.html

13. Knot Theory Online - The Web Site For Learning More About Mathematical Knot Theo KT HISTORY History of knot theory. This page is packed with fun and interesting diversionsfor you to enjoy as you learn more about the world of knot theory 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

14. Knot Theory References Books and articles about knot theory. The two best books. Livingston, Charles.(1993). knot theory. Washington, DC. Mathematical Association of America. http://www.cs.uidaho.edu/~casey931/new_knot/knbooks.html

Books and articles about Knot Theory The two best books Livingston, Charles. (1993). Knot Theory . Washington, DC.: Mathematical Association of America. ISBN: 0-83385-027-3 Adams, Colin C. (1994). The Knot Book: An Elementary Introduction to the Theory of Knots . New York: W.H. Freeman. ISBN: 0-7167-23929-X

15. Journal Of Knot Theory And Its Ramifications (JKTR) World Scientific. Connections between knot theory and other aspects of mathematics and natural science .Category Science Math Topology Journals......Journal of knot theory and Its Ramifications intended as a forum for new developmentsin knot theory, particularly developments that create connections between http://www.worldscinet.com/jktr/jktr.shtml

What's New New Journals Browse Journals Search ... Mathematics Journal of Knot Theory and Its Ramifications (JKTR) This Journal is intended as a forum for new developments in knot theory, particularly developments that create connections between knot theory and other aspects of mathematics and natural science. The stance is interdisciplinary due to the nature of the subject. More Feature Articles (Free Online Sample Issue) Vol. 11, No. 5 (August 2002) Click here to access the full text articles now. A Formula for the Casson Knot Invariant of a 2-Bridge Knot Y. Mizuma Quandle Knot Invariants are Quantum Knot Invariants M. Graña Component-Isotopy of Seifert Complexes T. Kadokami Higher Order Linking Numbers, Curvature and Holonomy The Mystery of the Brane Relation S. Garoufalidis A Transfer Matrix Approach to the Enumeration of Knots About the Uniqueness of the Kontsevich Integral Ch. Lescop A Matrix Invariant of Curves in S L. Zulli On Non-Simple Reflexive Links Unusual Formulae for the Euler Characteristic J. Roberts Planar Laces Tunnel Number One Knots Have Palindrome Presentations Some Forthcoming Papers Links with Surgery Yielding the 3-Sphere Masakazu Teragaito Quandle Knot Invariants are Quantum Knot Invariants MatÍas Graña On the Le-Murakami-Ohtsuki Invariant in Degree 2 for Several Classes of 3-Manifolds Catherine Gille The intersection of spheres in a sphere and a new geometric meaning of the Arf invariant Eiji Ogasa A Formula for the Casson Knot Invariant of a 2-Bridge Knot Yoko Mizuma Quandle Knot Invariants are Quantum Knot Invariants

16. B A Knot Theory Primer For an introduction to knot theory, click here. For a presentation of my work,click here. To proceed directly to the results of my work, click here. http://www.inst.bnl.gov/~wei/mypage.html

Why are knots important? 1) High Energy Theory Witten, 1989: Chern-Simons Topological Field Theories. Kreimer, 1994: Feynman Diagrams. 2) Quantum Gravity Rovelli, Smolin 1988: Exact physical quantum states of the gravitational field correspond one-to-one with knot and link classes. 3) Condensed Matter and Statistical Physics Baxter 1987: The Potts Model. 4) Low-Dimensional Topology Birman et al: Embeddings of one-dimensional into three-dimensional manifolds. 5) Chemistry Sumners, 1987: The study of the macromolecules. 6) Biology The study of the DNA. For an introduction to Knot Theory, click here . For a presentation of my work, click here . To proceed directly to the results of my work, click here . To review the table of contents, click here 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.

17. Knots On The Web (Peter Suber) The most comprehensive collection of knotting resources on the web. Sections on knot tying, mathematical Category Science Math Topology knot theory......Sections on knot tying, mathematical knot theory, knot art, knot discussionforums, knot software, knot videos, and knot books. knot theory. http://www.earlham.edu/~peters/knotlink.htm

Knots on the Web Artwork credits 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. My sixth section is on Knot Videos . If written instructions and still photos don't explain the intricacies of knotting well enough for you, try some of these videos (or some instructional

18. History Of Knot Theory Biographies of early knot theorists. Many early papers on knot theory (in pdf format) including papers Category Science Math Topology knot theory......HISTORY OF knot theory. This 2001. EARLY PAPERS ON knot theory. A.Cayley,On a problem of arrangements, Proc. Royal Soc. Edinburgh, Vol. http://www.maths.ed.ac.uk/~aar/knots/

HISTORY OF KNOT THEORY This home page is devoted to the history of knot theory, and is maintained by Jozef Przytycki and Andrew Ranicki. Our e-mail addresses are a.ranicki@edinburgh.ac.uk and przytyck@math.gwu.edu Please email to either of us any suggestions of additional material. BIOGRAPHIES OF EARLY KNOT THEORISTS Links to biographical entries in St. Andrews Mathematics History Archive M.Dehn P.Heegaard T.P.Kirkman J.B.Listing ... W.Wirtinger BIBLIOGRAPHY OF P.G.TAIT Provisional Bibliography prepared by Chris Pritchard, July, 2001 EARLY PAPERS ON KNOT THEORY A.Cayley, On a problem of arrangements, Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 338-342 Crum Brown, On a case of interlacing surfaces, Proc. Royal Soc. Edinburgh, Vol. 13, 121 (1885-6), 382-386 M.G.Haseman On knots, with a census of the amphicheirals with twelve crossings Trans. Roy. Soc. Edinburgh, 52 (1917-8), 235-255 also Ph.D thesis, Bryn Mawr College, 1918 M.G.Haseman Amphicheiral knots Trans. Roy. Soc. Edinburgh 52 (1919-20), 597-602. T.P.Kirkman

19. Jorge Pullin With Gambini we made several discoveries, including the fact that the celebratedJones polynomial of knot theory appeared to solve the quantum Einstein http://www.phys.lsu.edu/faculty/pullin/

Jorge Pullin Horace Hearne Chair in theoretical Physics, Louisiana State Adjunct Professor of Physics, University of Utah Adjunct Professor of Physics, PennState Ph.D., Instituto Balseiro Honors and awards Phone/Fax: (225)578-0464 pullin@phys.lsu.edu Want to hear those pipes? I recently joined the faculty at LSU, after being on the faculty at Penn State for eight years. This web page is in sort of transition between both places, with some links still pointing back to PSU. Research. Teaching. Service. Honors and awards. ... Background. Research My research interests cover many aspects of gravitational physics, both classical and quantum mechanical. I am currently focusing on two topics: quantum gravity and black hole collisions . You can also get my complete publication list . The explanations that follow are a bit longish, feel free to skip to the next topic if you get bored! Quantum gravity I collaborate with Rodolfo Gambini, of the University of the Republic in Montevideo, Uruguay, our collaboration has been going on since 1990. We coauthored a book "Loops, knots, gauge theories and quantum gravity"

20. Knot Description And Terminology Much of the knot theory is concerned with telling which knots are the sameand which of them are different. Some terms used in knot theory. http://www.mapleapps.com/categories/mathematics/Knot theory/html/knotdesc.htm

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