Geometry.Net - Math_Discover: Knot

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Starting with the flawed theory of Kelvin's knotted vortex to the work of Thurston, Jones and Witten, Category Science Math Topology knot theoryA Circular History of knot theory. It was through the study of 3 manifolds thatin the 1970's knot theory began returning to its ancestoral roots in physics.
http://www.math.buffalo.edu/~menasco/Knottheory.html

Extractions: 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
Liverpool Pure Maths: Knot Theory
knot theory Research Group.
http://www.liv.ac.uk/~su14/knotgroup.html
The Knot Theory MA3F2 Page
Includes examples, solutions, knot tables, pretty pictures. Course material includes colouring, Category Science Math Topology knot theoryThe knot theory MA3F2 page. Course material Prerequisites Little more thanlinear algebra plus an ability to visualise objects in 3dimensions.
http://www.maths.warwick.ac.uk/~bjs/MA3F2-page.html

Extractions: Course material Prerequisites Little more than linear algebra plus an ability to visualise objects in 3-dimensions. Some knowledge of groups given by generators and relations, and some basic topology would be helpful. The lectures and mind map that follow (from 2002) will be updated as we go through 2003. Mind map Course structure Lectures Writhe and linking numbers, Reidemeister moves and colouring, Colouring, Splittable links and chess boarding, Quadrilateral decomposition, Application of Cramer's rule, The determinant of a link, The colouring group, The number of colourings, Mirrors and codes, The Alexander polynomial, Knot sums, Bridge number and plats, Daisy chains and braids, Braids and Seifert circles, Alexander's theorem: links to braids, Seifert circles and trees, The bracket polynomial, The Jones polynomial, The skein relations, Alternating links, Span(V) = number of crossings, Tangles, Rational tangles and continued fractions, Tangled DNA, Genus and knot sum, Genus of a numerator
ThinkQuest
Elementary introduction to knot theory. Covers the existence of knots, Reidemeister moves and colorations.
http://library.thinkquest.org/12295/main.html
MA3F2 Knot Theory
MA3F2, Term 2. knot theory, 15 CATS. Commitment 30 lectures. Content Therehas been an explosion of interest in knot theory in the last ten years.
http://www.maths.warwick.ac.uk/undergrad/pydc/pink/pink-MA3F2.html

Extractions: Term 2 Knot Theory 15 CATS Status List A Prerequisites MA106 Linear Algebra MA130 From Geometry to Groups MA242 Algebra I and MA245 Algebra II would also be useful, as would MA3F1 Introduction to Topology Commitment : 30 lectures Content : There has been an explosion of interest in knot theory in the last ten years. Surprising connections with quantum physics, statistical mechanics and the action of enzymes on DNA have emerged. A knot may be regarded as a continuous loop of (thin rubber) string. There are two fundamental problems: Is the loop really knotted? When is a loop got from another by continuous deformation? The problem is tackled by computing invariants. If for instance we have a computable way to assign invariant numbers to knots then two knots with different numbers can not be equivalent. Another approach is to look at the topology of the complement of the knot. Can we find a surface with the knot as boundary? What properties does it have? Aims : To introduce a variety of ways of representing knots and explore the value and novelty of different approaches.
The Mathematics And Origin Of String Figures
Trivial knot theory, history, and a few new designs by the author, Martin Probert.Category Arts Crafts String Figures1 String Figures and knot theory. 2 The origin of string figures. StringFigures and knot theory mathematics of the unknot under tension.
http://website.lineone.net/~m.p/sf/menu.html

Extractions: Includes construction methods for four ethnographic string figures. Many ethnographical string figures have come down to us only in the form of an ambiguous photograph in which, at the crossings, it is impossible to determine by eye which string lies over which. A knowledge of the set of all look-alikes ('similar-looking string figures') is a considerable aid in attempting to reconstruct such a figure. Part I shows how such a set may be determined mathematically (using the techniques of knot theory). Part III shows how such a set may be obtained manipulatively . Several conjectures are made concerning this manipulative process on the set of all look-alikes.
Knot Theory -- From MathWorld
An overview of knot theory from Mathworld
http://mathworld.wolfram.com/topics/KnotTheory.html
String Figures And Knot Theory: Part I
String Figures and knot theory mathematics of the unknot under tension.by Martin Probert. This is quite different from the knots of knot theory.
http://website.lineone.net/~m.p/sf/sfmaths1.html

Extractions: - mathematics of the unknot under tension by Martin Probert First posted February 2001. Last revised December 2002. Many ethnographical string figures have come down to us only in the form of an ambiguous photograph in which, at the crossings, it is impossible to determine by eye which string lies over which. A knowledge of the set of all look-alikes ('similar-looking string figures') is a considerable aid in attempting to reconstruct such a figure. Part I shows how the set of all look-alikes may be determined mathematically : certain subsets of string contacts are labelled, the figure is then projected onto two dimensions, and finally the techniques of knot theory can be applied to identify the look-alikes. Part II contains examples illustrating the results. Part III shows how the set of all look-alikes may be obtained manipulatively : a process of 'unravelling by motifs' is introduced which transforms one look-alike into another. Part IV contains an analysis of a number of string figures. Part V contains several conjectures concerning the unravelling process on the set of all look-alikes.
Knot Theory
knot theory. The For example, my current favorite is The Knot Book byColin Adams. This is listed in a bibilography of knot theory. The
http://grail.cba.csuohio.edu/~somos/knots.html

Extractions: The use of knots goes back to pre-history, but the mathematical study of knots goes back only to the 19th century. Some of the early investigators were Gauss, Listing, Kirkman, Tait, and Little. Due to connections with applied fields like physics and biology there is increased research today. For examples look at the Titles in Series on Knots and Everything edited by Louis H. Kauffman . There are books which explain the topic for a more popular audience. For example, my current favorite is The Knot Book by Colin Adams . This is listed in a bibilography of knot theory . The story behind Making a Mathematical Exhbition by Ronald Brown and Tim Porter describes some of the issues involved in presenting knot theory. There are web sites which you can visit to find out more. You can start with the Mega-Math section on knots. Then switch to the KnotPlot site for great color graphics. A less ambitious site is Geometry and the Imagination section on knot notation. Knots on the Web by Peter Suber has a long well-annotated list of links on knot theory as well as many other aspects of knots. Mouse Bousfield has a nice Knot Theory site.
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.
http://www.maths.ed.ac.uk/~aar/knots/index.htm

Extractions: Please email to either of us any suggestions of additional material. Links to biographical entries in St. Andrews Mathematics History Archive 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
Knot Theory
Knots whose ends were glued together and their classification formthe subject of a branch of Topology known as the knot theory.
http://www.cut-the-knot.com/do_you_know/knots.shtml

Extractions: Recommend this site Knots... Every one knows from experience how to create a knot. We do this all the time, often unwittingly. Knots whose ends were glued together and their classification form the subject of a branch of Topology known as the Knot Theory. On the left there is a picture of the Left Trefoil knot. On the right there is the Right Trefoil knot. It's impossible to continuously (i.e. stretching and twisting but without causing damage to either of them) deform one into another. However, it must be noted that the two knots are topologically equivalent in the sense that there exists a topological transformation that maps one into another. The knots are mirror reflections of each other. In the real world, it can be argued that mirror reflections are only mental images whose existence is entirely different from that of the objects whose reflections they are. In Mathematics, reflections are as real as the objects themselves. Mathematically, reflections are topological transformations that could not be carried out on the real world objects. But
Knot Theory On WWW
knot theory on WWW. Japanese Books on knot theory. ?. ?, RH.?, RH.?
http://home.hiroshima-u.ac.jp/teragai/knot.html

Extractions: Japanese Books on Knot Theory uÑÚ_üåv uÑÚ_Tàv C W.B.R. RbV()CºãÄ¼óCVvK[tFA[NC2000N uÑÚÌwv uÑÚ_Æ»Ìpv uÑÚ_v C Íà¾v(Ò)CVvK[tFA[NC1990N uÑÚ_üåv C éØWê()CTCGXÐC1991N uÑÚÌwÆ¨v C ¬ÑêÍ()C©qXC1992N uRs[^ÉæéÑÚ_üåv C LsCRcCiCLcpüq()CqìXC1996N uR³½lÌüåv uÑÚÆÊqQv C ºã()C©qXC2000N uÊqsÏÊv uü`ã©çzW[Öv uUCNÆR³½lÌv C J.M. eVmX()COc(ó)CVvK[tFA[NC1992N
Knot Theory On WWW
Home, knot theory on WWW, Links to Personal Pages, Profile. knot theory on WWW. KnotTheory Group Articles and Preprints; Links to webpages related to knot theory;
http://home.hiroshima-u.ac.jp/~teragai/knot-e.html

Extractions: Topology in Matsue 2002 (June 24-28,2002, Shimane University) The 9th Korea-Japan School of Knots and Links (Jan 14-17,2002, Nara Women's University) Conference "Knot theory, Now-Past-Future" (Oct 22-25, 2001, Waseda University) Conference "Topology of Knots IV" (Dec 17-20,2001, Osaka City University) ... Journal of Knot Theory and its Ramifications
Links To Low-dimensional Topology: Knot Theory
General Conferences Pages of Links knot theory 3manifolds Miscellany. KnotTheory. The page of the knot theory Group at the Univ. of Liverpool.
http://www.math.unl.edu/~mbritten/ldt/knots.html

Extractions: General Conferences Pages of Links 3-manifolds ... Home pages Joe Christy has put together www.computop.org , to serve as a source for the computational 3-dimensional topology community. The site includes links to downloadable software, and a set of mailing lists. The page of the Knot Theory Group at the Univ. of Liverpool. An introduction to knot theory which seems to be aimed at teachers of mathematics can be found at Los Alamos National Laboratory There is also another knot theory page at the University of British Columbia. Another page , developed from a course for liberal arts students, is at York Univ. A discussion, and several lists, concerning the classification of knots, may be found in Charilaos Aneziris' home page. This table of knots up to nine crossings came from Sean Collom 's home page at Oxford. A collection of pages on Mathematics and Knots at the University of Wales. A huge page of links to pages on knots and knot theory of all kinds. An on knot theory appears in the November 1997 issue of American Scientist A page at the Univ. of Liverpool for accessing preprints on knot theory.
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.
http://www.cs.ubc.ca/labs/imager/contributions/scharein/KnotPlot.html

Extractions: 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. 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. 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)
Knot-Theory.com - Those Who Can't Do, Theorize.
Yo-Yo Trick Library including written description and downloadable videos.Category Recreation Collecting Toys Yo-Yos TricksWelcome to the knot theory Skill Toy Trick Library Most Viewed 1. Dirty Bomb.2. Kamikaze. 3. Aluminum Whip. 4. And Whut? 5. AAA Zipper. 6. Dirty Mount. 7. A1.
http://www.knot-theory.com/tricklib/index.php

Extractions: Dirty Bomb Kamikaze Aluminum Whip And Whut? ... Abomination This database serves as a central index for skill toy trick descriptions, illustrations, videos, and other information, contained both on this page and on others around the net. The main purpose is the spread information quickly and easily about tricks both old and new, to help new players learn, or experienced performers teach.
Mathematics And Knots Exhibition
High school level introduction to knot theory. Covers colourings, connected sums, torus knots, prime knots and applications of knot theory.
http://www.bangor.ac.uk/cpm/exhib/

Extractions: This material may be used freely for educational, artistic and scientific purposes, with acknowledgement, but may not be used for commercial purposes, for profit or in texts without the permission of the publishers. Link to Review of the Exhibition Borrowing the exhibition Why study Mathematics? Studying mathematics at Bangor ... Knots on the web
The Math Forum - Math Library - Knot Theory
mathematics. This page contains sites relating to knot theory. Browse andSearch the Library Home Math Topics Topology knot theory.
http://mathforum.org/library/topics/knot_theory/

Extractions: A collection of knots and links, viewed from a partly mathematical perspective. Images on this site were created with KnotPlot, a program designed to visualize and manipulate mathematical knots in three and four dimensions. A picture gallery, description of the features of the program, and links to other relevant sites on the Web are included. more>> Knots on the Web - Peter Suber, Earlham College A comprehensive list of knot resources on the Web, annotated and organized into three categories: knot tying, knot theory, and knot art. Also Knot books and a Knots Gallery displaying images from the newsgroup rec.crafts.knots. more>> Knot Theory (The Geometry Junkyard) - David Eppstein, Theory Group, ICS, UC Irvine
Math Forum - Ask Dr. Math
Drexel dragon Donate to the Math Forum Associated Topics Dr. Math Home Search Dr. Math What is knot theory? Question What is knot theory?
http://mathforum.org/library/drmath/view/51653.html

Extractions: Associated Topics Dr. Math Home Search Dr. Math Date: 03/10/98 at 23:46:23 From: keatha Subject: the knot theory Dear Dr. Math, In order for me to complete my project on knot theory, I need a question, hypothesis, and a purpose. I have this information, I just want you to proofread it to see if I need to add anything or delete anything. Question: What is Knot Theory? Purpose: To prove that knot theory is related to topology. Hypothesis: If topology deals with the bending of objects, then it (knot theory) is related to topology. Thanks a million, Keatha Date: 03/11/98 at 10:32:24 From: Doctor Sonya Subject: Re: the knot theory Dear Keatha, I just wrote a big paper on knot theory. What a great topic to choose. Your question is good, and your hypothesis is correct that topology deals with the bending of objects, and that topology is related to knot theory. However, if you really want to answer the question, "What is knot theory?" I don't think, "Knot theory is related to topology." is enough of an answer. When you get to the real details of it, knot theory is also closely related to modern algebra (although you'd never know it from all the pretty pictures of knots!). The most exciting thing about knot theory that deals with the bending and twisting of knots is something called the "Jones Polynomial," discovered by a mathematician named Jones and written about by Louis Kauffman. If you still have to do more research, this might be interesting for you to include. Also, take a look at "The Knot Theory Home Page"
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).
http://www.ams.org/notices/199505/sumners.pdf

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