61. Professor Lomonaco: Five Dimensional Knot Theory Five Dimensional knot theory. by. Samuel J. Lomonaco, Jr. (*) Publishedin Low Dimensional Topology, Contemporary Mathematics Series http://www.cs.umbc.edu/~lomonaco/Abstract4.html

Five Dimensional Knot Theory by Samuel J. Lomonaco, Jr. Published in "Low Dimensional Topology," Contemporary Mathematics Series of the American Mathematics Society, Providence, Rhode Island, Vol. 20 (1984), pp 249 - 270. ABSTRACT. This paper outlines in an informal and intuitive fashion a method for computing the second homotopy group p (X) (as a left Z p (X) -module) of the complement X = S - kM of a smooth -knot (S , kM . By a smooth -knot (S , kM is meant a smooth imbedding of a closed -manifold M in the -sphere S . An example of the calculation is given. The methods of this paper may also be used to compute the Rohlin invariant of -manifolds from their imbeddings in S The paper (including figures) in postscript form (11.6MB) (29 pages) The paper (including figures) in gzipped postscipt (890KB) (29 pages) View Figures in HTML Cartoon based on this paper (*) Partially supported by the L-O-O-P Fund.

62. Knot Theory Resources knot theory resources. Recommended References. see index for totalcategory for your convenience Best Retirement Spots Teacher http://futuresedge.org/mathematics/Knot_Theory.html

Knot Theory resources. Recommended References. [see index for total category] for your convenience: Best Retirement Spots Web Hosting ULTRAToolBox Resources on Diet and Nutrition Pain Relief Allergies Tech Refresh , and finally - a must check - Mediterranean diet Discovery. Knot Theory applications, theory, research, exams, history, handbooks and much more Introduction: Knot Book: An Elementary Introduction to Mathematical Theory of Knots by Colin C. Adams Introduction to Knot Theory by H. R. Crowell The Mathematical Theory of Knots and Braids: An Introduction by Siegfried Moran An Introduction to Knot Theory (Graduate Texts in Mathematics, 175) by W. B. Raymond Lickorish The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots by Colin C. Adams Applications: Theory: Knot Theory (Carus Mathematical Monographs, No 24) by Charles Livingston Knot Book: An Elementary Introduction to Mathematical Theory of Knots by Colin C. Adams Gauge Fields, Knots, and Gravity (Series on Knots and Everything, Vol. 4) by John C. Baez On Knots. (AM-115)

63. KNOT THEORY LINKS Galois Relations on Knot Invariants, by T. Gannon and MA Walton, 95/09;Noncommutative Geometry of Finite Groups, by K. Bresser et al., 95/09; http://web.mit.edu/afs/athena.mit.edu/user/r/e/redingtn/www/netadv/knots.html

KNOTS AND BRAIDS (AND) LINKS Galois Relations on Knot Invariants, by T. Gannon and M. A. Walton, 95/09 Noncommutative Geometry of Finite Groups, by K. Bresser et al., 95/09 The Many-Anyon Problem, by Claus Montonen, 95/02 Braided Geometry and q-Minkowski Space, by Shahn Majid, 94/10 ... Spin networks, by John C. Baez To contribute to this page, write Norman Redington, redingtn@mit.edu RETURN TO THEORETICAL PHYSICS LINKS RETURN TO NET ADVANCE OF PHYSICS HOMEPAGE

64. Dror Bar-Natan:Classes:2000-01:Knot Theory Dror BarNatan Classes 2000-01 knot theory. Instructor Dror Bar-Natan, drorbn@math.huji.ac.il,02-658-4187. Classes Tuesdays 1400-1600 at Sprintzak 215. http://www.math.toronto.edu/~drorbn/classes/0001/KnotTheory/

Dror Bar-Natan Classes Knot Theory Instructor: Dror Bar-Natan drorbn@math.huji.ac.il Classes: Tuesdays 14:00-16:00 at Sprintzak 215. Office hours: Sundays 14:00-15:00 in my office, Mathematics 309. Agenda: To learn about finite type knot invariants, and especially about their multiply-proven but not-sufficiently-well understood fundamental theorem , whose different proofs relate to almost everything in mathematics. Further resources: Advertisement Poster (with some extra syllabus details). Pathologies - strange things happen in the continuous category. A very simple Mathematica program that can verify the invariance of the Kauffman bracket and compute it on small knots: Mathematica Notebook (requires Mathematica or MathReader PDF (requires Acrobat Reader or GhostView A handout on "Some Dimensions of A": ChordDiagrams.ps The files needed to make ChordDiagrams.ps: ChordDiagrams.tar.gz Class notes for April 17th, 2001 . (Reminders, the algebra structure of , algebras and co-algebras, the co-algebra structure of Class notes for April 24th, 2001

65. Dror Bar-Natan: Classes: 2001-02: Knot Theory Seminar Dror BarNatan Classes 2001-02 Seminar on knot theory. Agenda Have everystudent give at least one fun lecture on elementary knot theory. http://www.math.toronto.edu/~drorbn/classes/0102/KnotTheory/

Dror Bar-Natan Classes Seminar on Knot Theory Instructor: Dror Bar-Natan drorbn@math.huji.ac.il Meetings: Mondays 12:00-14:00 at Upper Papik. Office hours: Tuesdays 14:00-15:00 in my office, Mathematics 309. Agenda: Have every student give at least one fun lecture on elementary knot theory. Prerequisites: Meant for advanced undergraduate students. Reading material: W.B. Raymond Lickorish An Introduction to Knot Theory , GTM 175, Springer-Verlag, New York 1997. The books by Kauffman and Rolfsen V. V. Prasolov and A. B. Sossinsky's Knots, links, braids and 3-manifolds: an introduction to the new invariants in low-dimensional topology, Translations of Mathematical Monographs 154, American Mathematical Society 1997. Thanks, Raz , for the tushim! Some suggested topics for student lectures: (more may be added later, and you may choose subjects not on this list!) Topic Speaker and Date Details Dependencies Reidemeister's theorem Lior Zaibel, March 18th Prove that any two diagrams for the same knot are connected by a sequence of Reidemeister moves. See almost any book on knot theory. None.

66. Knot Theory knot theory. knot theory studies the placement of onedimensionalobjects called strings 23,24,25 in a three-dimensional space. http://www.drchaos.net/drchaos/Book/node141.html

Next: Crossing Convention Up: Knots and Templates Previous: Example: Duffing Equation Knot Theory Knot theory studies the placement of one-dimensional objects called strings [23,24,25] in a three-dimensional space. Figure 5.6: Planar diagrams of knots: (a) the trivial or unknot ; (b) figure-eight knot ; (c) left-handed trefoil ; (d) right-handed trefoil; (e) square knot ; (f) granny knot. A simple and accurate picture of a knot is formed by taking a rope and splicing the ends together to form a closed curve. A mathematician's knot is a non-self-intersecting smooth closed curve (a string ) embedded in three-space. A two-dimensional planar diagram of a knot is easy to draw. As illustrated in Figure , we can project a knot onto a plane using a solid (broken) line to indicate an overcross (undercross). A collection of knots is called a link (Fig. Figure 5.7: Link diagrams: (a) Hopf link ; (b) Borromean rings ; (c) Whitehead link The same knot can be placed in space and drawn in planar diagram in an infinite number of different ways. The equivalence of two different presentations of the same knot is usually very difficult to see. Classification of knots and links is a fundamental problem in topology. Given two separate knots or links we would like to determine when two knots are the same or different. Two knots (or links) are said to be

67. Knot Theory RAP knot theory RAP. When, Th 200350 (with a break). http://www.math.uiuc.edu/~brinkman/teaching/rap/

Knot Theory RAP When Th 2:00-3:50 (with a break) Where Altgeld Organizers Peter Brinkmann and Nadya Shirokova Email brinkman@math.uiuc.edu or nadya@math.uiuc.edu This RAP serves two purposes. First, we will discuss some of the basics of knot theory such as knot projections, types of knots, knot groups, etc., loosely based on some introductory texts (such as Adams, 'The Knot Book'; Crowell-Fox, 'Introduction to Knot Theory'; Burde-Zieschang, 'Knots'; Rolfsen, 'Knots and Links'; Kauffman, 'On Knots'). Second, the RAP will serve as a platform for talks about current research (such as Vassiliev invariants, geometric knot theory, and slalom knots). Archive: Fall 01 Spring 02 Peter Brinkmann $Date: 2002-06-03 01:33:18-05 $

68. Knot Theory RAP knot theory RAP. When, Th 200250. 09/06/2001, Peter Brinkmann, Introductionto knot theory. 09/13/2001, Peter Brinkmann, Introduction to knot theory (cont.). http://www.math.uiuc.edu/~brinkman/teaching/rap/fall01.html

Knot Theory RAP When Th 2:00-2:50 Where Altgeld Organizer Peter Brinkmann Office Altgeld Email brinkman@math.uiuc.edu This RAP serves two purposes. First, we will discuss some of the basics of knot theory such as knot projections, types of knots, knot groups, etc., loosely based on some introductory texts (such as Adams, 'The Knot Book'; Crowell-Fox, 'Introduction to Knot Theory'; Burde-Zieschang, 'Knots'; Rolfsen, 'Knots and Links'; Kauffman, 'On Knots'). Second, the RAP will serve as a platform for talks about current research (such as Vassiliev invariants, geometric knot theory, and slalom knots). Please let me know if you would like to give a talk. Date Speaker Title Organizational meeting Peter Brinkmann Introduction to knot theory Peter Brinkmann Introduction to knot theory (cont.) Nadya Shirokova Finite type knot invariants We will discuss the axiomatics for the invariants of finite type, introduced by V.Vassiliev. We will study their properties and show that classical invariants, like Alexander-Conway polynomial can be decomposed over invariants of finite type. Katharine Preedy Types of knots Nadya Shirokova Finite type knot invariants (cont.)

69. Knot Theory And Fullerenes 3. knot theory and fullerenes Let be given an oriented knot or linkdiagram D with generators g 1 , , g n 12. If the generators http://members.tripod.com/~modularity/ful4.htm

3. Knot theory and fullerenes Let be given an oriented knot or link diagram D with generators g g n ]. If the generators g i g j g k are related as in Fig. 6a , then a ii t a ij a ik = -1; if they are related as in Fig. 6b , then a ii t a ij a ik = -1; in all the other cases a ij =0. The determinant d t a ij D . In the case of links, in this polynomial invariant we use for generators of different components different variables (up to a permutation of variables). For example, let us show that two isomers of C are different (Fig. ). After converting their chemical Schlegel diagrams into the alternating knot diagrams, denoting their generators, and calculating the corresponding determinants we obtain, respectively, D t t t t t t t t t t and D t t t t t t t t t t , proving their difference. On the other hand, for three-component alternating links corresponding to the diagrams ( Fig. 6d ), using the same multivariable invariant for link projections, we conclude that they both represent the isomorphic Schlegel diagrams of the same fullerene C Figure 6. NEXT CONTENTS

70. A Family Of Impossible Figures Studied By Knot Theory A family of impossible figures studied by knot theory Corinne Cerf. MathematicsDept., CP 216. Université Libre de Bruxelles. B1050 Bruxelles, Belgium. http://members.tripod.com/vismath8/cerf/

A family of impossible figures studied by knot theory Corinne Cerf Mathematics Dept., CP 216 B-1050 Bruxelles, Belgium ccerf@ulb.ac.be 1. Introduction Impossible figures are fascinating objects, related to art, psychology, and mathematics [ ]. Lionel and Roger Penrose (father and son) introduced the impossible tribar in 1958 [ Fig. 1 ). A figure is called impossible when "a contradiction in our interpretation is noticed but does not result in our rejecting it in favour of a consistent one" [ ]. The object represented in Fig. 1 is an impossible figure because our mind tries to interpret it as a three-dimensional (3D) object in the Euclidean space, with straight edges and planar faces, instead of interpreting it, for example, as a two-dimensional object drawn on the paper plane (which is perfectly possible). Fig. 1 Impossible figures have inspired researchers with more than one hundred papers (see Kulpa [ ] for an extensive bibliography), and the Dutch artist Escher [ ] with some famous drawings (see e.g. Fig. 2

71. Knot Theory knot theory. Computer prograoms for knot theory (Lexicographical) Knotscape (Linux,SUN); Kodama knot (Linux, SUN); Mr.Monomie (Macintosh); OPTi (Macintosh); http://www.econ.ryukoku.ac.jp/~nakagawa/math/knots/e-index.html

Knot theory [Mathematics] [Museum] [Japanese] Results of my study : Mathematics A family of integer-valued complete invariants of oriented knot types Unknotting tunnels of Montesinos links Shortest vertical geodesics of manifolds obtained by hyperbolic Dehn surgery on the Whitehead link (with H.AKIYOSHI and M.SAKUMA) Results of my study : others KnoTeX ver 2.0, i.e. LaTeX2e macros for Knot theorists Links for links Communications in NKOOK seminar Computer prograoms for Knot theory (Lexicographical) Knotscape (Linux, SUN) Kodama knot (Linux, SUN) Mr.Monomie (Macintosh) OPTi (Macintosh) Snappea (Linux, Macintosh, MS-Windows) Teruaki (Macintosh, Sharewere) Please teach me by e-mail if you know other computer programs for Knot theory.

72. Knot Theory By Kurt Reidemeister Topology Atlas Book Abstract iaad16 © Copyright by BCS Associates KnotTheory by Kurt Reidemeister, ISBN 0-914351-00-1 Order this book from BCS! http://at.yorku.ca/i/a/a/d/16.htm

Topology Atlas Book Abstract # iaad-16 BCS Associates Knot Theory by Kurt Reidemeister ISBN 0-914351-00-1 Order this book from BCS! An English translation of Springer-Verlag's 1932 German edition. Contents Foreword to the English edition Publisher's foreword to the original edition Introduction Chapter I: - Knots and their projections Definition of a knot Regular projections The subdivision of the projection plane into regions Normal knot projections Braids Knots and braids Parallel knots, Cable knots Chapter II: - Knots and matrices Elementary invariants The matrices (c h The matrix (a i The determinant of a knot The invariance of the trosion numbers The torsion numbers of particular knots The quadratic form of a knot Minkowski's units Minkowski's units for particular knots A determinant inequality Classification of alternating knots Almost alternating knots Almost alternating circles The L-polynomial of a knot L-polynomials of particular knots Chapter III: - Knots and Groups Equivalence of braids The braid group Definition of the group of a knot Invariance of the knot group The group of the inverse knot and of the mirror image knot The matrix (l ik x)) and the group The knot group and the matrices (c h The edge path group of a knot Structure of the edge path group Covering spaces of the complementary space of the knot The group of a parallel knot The groups of torus knots The L-polynomials of parallel knots Several special knot groups A particular covering space Table of knots Bibliography Index BCS Associates has given its consent to include this document in

73. TOPCOM, Book Review Of Knot Theory By Corinne Cerf Topology Atlas Document topc85 Production Editor Thomas M. ZachariahA Book Review knot theory. knot theory by K. Reidemeister. http://at.yorku.ca/t/o/p/c/85.htm

Topology Atlas Document # topc-85 A Book Review: Knot Theory by Corinne Cerf Mathematics Department, CP 216, Universite Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium Book Review from Volume 4, #2 , of TopCom Knot Theory by K. Reidemeister. Originally published as Knotentheorie by K. Reidemeister, Ergebnisse der Mathematik und ihrer Grenzgebiete, Alte Folge, Band 1, Heft 1, SPRINGER, Berlin, 1932. Translated from the German and edited by L. F. Boron, C. O. Christenson, and B. A. Smith, BCS Associates , Moscow, Idaho, 1983 . xv+143 pp. ISBN 0-914351-00-1 This book is a 1983 translation of the 1932 celebrated book by Kurt Reidemeister. It is subdivided into three chapters. The first one is an introduction to knots and braids, including (a sketch of) the original proof that two knots are equivalent if and only if their projections are related by a finite sequence of the three so-called Reidemeister moves. The second chapter describes the main knot invariants obtainable from matrices, like linking numbers, torsion numbers, determinants, and L-polynomials, now called normalized Alexander polynomials, that have been discovered independently by Reidemeister and Alexander. The third chapter deals with knot groups: definition by generators and relations from a projection, invariance, equivalence with the fundamental group of the knot complement, calculation of the group of special families of knots. A group-theoretic interpretation of the matrices and L-polynomials of Chapter II is given.

74. Knot Theory 2 July 2001. knot theory and Fluid Mechanics A Reflection on the Workof Tait and Kelvin. Professor Moffat described several recent http://www.ma.hw.ac.uk/RSE/meetings_etc/ordmtgs/2001/reports/knots.htm

Professor Keith Moffat Director, Isaac Newton Institute for Mathematical Sciences University of Cambridge 2 July 2001 Knot Theory and Fluid Mechanics A Reflection on the Work of Tait and Kelvin Unable to reproduce Tait’s smoke rings (due to lecture room smoke-detectors), Professor Moffat demonstrated the creation of vortex rings in water using coloured dyes and went on to show how more complex interactions can occur. He noted that Kelvin’s interest in the subject was motivated by his desire to develop an atomic theory in terms of vortices "in the aether". Kelvin recognised that knotted vortex filaments (or tubes of very small cross-section) would permanently retain their knotted form, and that the characteristic frequencies of oscillations of these forms about their equilibrium states (if such existed) might provide a fundamental explanation for the known spectroscopic properties of matter. Professor Moffat concluded his lecture with descriptions of more recent mathematical studies in this area, including his own work on the concept of helicity.

75. Knot Theory I work with Dr. Eric Rawdon of the math department on physical knottheory. He has created a computational model for polygonal knots http://www.math.duq.edu/~piatek/research/knot.htm

About Research Academics Photos Fun Math Home ... CS Home I work with Dr. Eric Rawdon of the math department on physical knot theory. He has created a computational model for polygonal knots in order to minimize rope length and other energies. My work involves developing parallel systems, algorithms, and other methods to speed up the minimization process. Background Knot theory is a branch of topology , that deals with the properties of mathematical knots . Those mathworld definitions, although precise, provide more information than is needed to understand our work. Fundamentally, a mathematical knot is identical to the everyday understanding of knots, with the exception that mathematical knots are closed curves (read: without ends) that have no thickness. However, our work centers around physical knots that do have thickness. Goals Our goal is to minimize rope length and other energies (ways of 'rating' knots) computationally. This generally makes the knots much more recognizable (see picture at left), but it can require a vast amount of computing time, especially for knots with a lot of edges. So, an associated goal is finding ways of speeding up the minimization task. * More in-depth motivation for energy minimization is available on Dr. Rawdon's

76. Really Bad Knot Theory Puns These terrible puns are the output of the knot theory Mini course I cotaught atthe 2000 Hampshire College Summer Studies in Mathematics with Emily Peters (UC http://cerebro.cs.xu.edu/~smbelcas/knotpuns.html

These terrible puns are the output of the Knot Theory Mini course I co-taught at the 2000 Hampshire College Summer Studies in Mathematics with Emily Peters (UC). Read aloud for maximal effect. All that Knot Theory for Naught! Tie him (Wing Mui) into a Wing Knot. We must define a knot, because if we do not, then we do not know what is a knot and what is not a knot. John "Still Chicken" Choi: "Called the 'trefoil' knot. Let's say it again, the 'trefoil' knot. ...Called the 'figure 8' knot... let's knot say it again..." Amanda Redlich: "Knots to you." "She's knots. We're knot able to..." Wing: "Knot Theory Chapter 4. Making a knot not a knot, or an unknot, or not... knot?" "2 types of knots? knot!" John Darius Mangual: "...in knot theory we may be knot sure or not sure." John Basias: "Our not rules for not polynomials." Emily: "Knotting my knitting; I mean not knotting; I mean infinknitting; I mean making bad puns." Wing: "Not cool? That is not true for all knot theorists not working on knotting knots but not slacking in their studies in the theory of knots not for naught." No, These Are Knot Puns

77. Geometry Topology Mathematics Knot Theory And Its Applications Kunio Murasugi Geometry Topology Mathematics knot theory and Its Applications Kunio Murasugi. SubjectGeometry Topology Mathematics Title knot theory and Its Applications http://www.book-planet.co.uk/Kunio-Murasugi-Knot-Theory-and-Its-Appli-3764338172

Geometry Topology Mathematics Knot Theory and Its Applications Kunio Murasugi Subject: Geometry Topology Mathematics Title: Knot Theory and Its Applications Author: Kunio Murasugi Arthur Kirchhofer Conservation... Ove Franzen Somesthesis and th... I Kuzin Entire Solutions of Se... Charles Broto Algebraic Topolo... ... Bartkowski Oliver DVD Guide d...

78. Problems In Knot Theory Problems in knot theory. There are already several collections of problemsin knot theory available. So this is mainly a collection http://math.ucr.edu/~xl/knotprob/knotprob.html

Problems in Knot Theory There are already several collections of problems in knot theory available. So this is mainly a collection of problems of personal interest. Feel free to send me any comments you might have: maybe the answer is obvious, maybe the answer exists already in literature, maybe the problem should be attributed more appropriately, etc. Also, it is welcome if anyone would like to suggest some problems to this collection. Nevertheless, since this is a personal collection, I reserve the right to decide whether to put the suggested problems into this collection according to my own taste. Surgery modification is a procedure of modifying a link in or by performing a Dehn surgery on an unknotted circle having linking number zero with all components of the link in question. Surgery equivalence is then the equivalence relation generated by surgery modification. The classification of links up to surgery equivalence is done by J. Levine (Topology,1987). We may refine the notion of surgery modification as follows. For simplicity, let us consider only links with vanishing linking numbers. If we assume further that the unknotted circle used to perform a surgery modification has vanishing Milnor's triple linking numbers with other componenets of the given link, we call such a surgery modification of "second order". It can be shown that the Sato-Levine invariant is invariant under surgery modification of second order. Classify links with zero linking numbers up to surgery equivalence of second order.

79. Knot Theory - Wikipedia knot theory. From Wikipedia, the free encyclopedia. knot theory isa recently discovered. An introduction to knot theory. Given a http://www.wikipedia.org/wiki/Knot_theory

80. Knot Theory History knot theory. History. This Johann Frederich Carl Gauss (17751855).The history of knot theory began in the very early 1800's. Johann http://www.kpbsd.k12.ak.us/kchs/JimDavis/CalculusWeb/Knot Theory History.htm

Knot Theory History This web site is for the basic understanding of where Knot Theory came from. Johann Frederich Carl Gauss (1775-1855) The history of Knot Theory began in the very early 1800's. Johann Frederich Carl Gauss was a German mathematician who was interested in the idea of knots. His contribution to the Knot Theory was "analysis situs." Analysis situs deals with the mathematical differences between the simple and complex knots. The mathematical differences are very different to the human eye, but until then, they were difficult to distinguish them mathematically. Lord Kelvin, William Thomson (1824-1907) Until the late 1800's no one was really interested in the idea of knots. Lord Kelvin, also known as William Thomson, was an English scientist who taught at the University of Glasgow. One of his greatest achievements was the idea of "ether." His belief was that the universe was filled with an invisible and frictionless fluid called "ether," and the atoms were the vortices in this fluid in the shape of knots. Thus a table of knots would be a table of elements. This of course was later on disproved when atomic ideas began to emerge. The Knot Theory soon faded away until the 1950's. James Watson (left) and Francis Crick (right) The race to discover the shape of deoxyribonucleic acid (DNA) was all over the world. When it was finally finish, the idea of knots came back to life. In 1953, James Watson and Francis Crick discovered that DNA was in the shape of a double helix. This concept led to discovering that DNA had parts of it knotted in many locations. With the help of both of these scientists, Knot Theory came back with scientist thinking up new ideas fighting diseases and understanding the make-up of all living things.

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