Computing the Writhe of a Knot
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The Borromean Rings: a Video About the New IMU Logo
The Borromean Rings: A Video about the New IMU Logo Charles Gunn and John M. Sullivan∗ Technische Universitat¨ Berlin Institut fur¨ Mathematik, MA 3–2 Str. des 17. Juni 136 10623 Berlin, Germany Email: {gunn,sullivan}@math.tu-berlin.de Abstract This paper describes our video The Borromean Rings: A new logo for the IMU, which was premiered at the opening ceremony of the last International Congress. The video explains some of the mathematics behind the logo of the In- ternational Mathematical Union, which is based on the tight configuration of the Borromean rings. This configuration has pyritohedral symmetry, so the video includes an exploration of this interesting symmetry group. Figure 1: The IMU logo depicts the tight config- Figure 2: A typical diagram for the Borromean uration of the Borromean rings. Its symmetry is rings uses three round circles, with alternating pyritohedral, as defined in Section 3. crossings. In the upper corners are diagrams for two other three-component Brunnian links. 1 The IMU Logo and the Borromean Rings In 2004, the International Mathematical Union (IMU), which had never had a logo, announced a competition to design one. The winning entry, shown in Figure 1, was designed by one of us (Sullivan) with help from Nancy Wrinkle. It depicts the Borromean rings, not in the usual diagram (Figure 2) but instead in their tight configuration, the shape they have when tied tight in thick rope. This IMU logo was unveiled at the opening ceremony of the International Congress of Mathematicians (ICM 2006) in Madrid. We were invited to produce a short video [10] about some of the mathematics behind the logo; it was shown at the opening and closing ceremonies, and can be viewed at www.isama.org/jms/Videos/imu/. -
Complete Invariant Graphs of Alternating Knots
Complete invariant graphs of alternating knots Christian Soulié First submission: April 2004 (revision 1) Abstract : Chord diagrams and related enlacement graphs of alternating knots are enhanced to obtain complete invariant graphs including chirality detection. Moreover, the equivalence by common enlacement graph is specified and the neighborhood graph is defined for general purpose and for special application to the knots. I - Introduction : Chord diagrams are enhanced to integrate the state sum of all flype moves and then produce an invariant graph for alternating knots. By adding local writhe attribute to these graphs, chiral types of knots are distinguished. The resulting chord-weighted graph is a complete invariant of alternating knots. Condensed chord diagrams and condensed enlacement graphs are introduced and a new type of graph of general purpose is defined : the neighborhood graph. The enlacement graph is enriched by local writhe and chord orientation. Hence this enhanced graph distinguishes mutant alternating knots. As invariant by flype it is also invariant for all alternating knots. The equivalence class of knots with the same enlacement graph is fully specified and extended mutation with flype of tangles is defined. On this way, two enhanced graphs are proposed as complete invariants of alternating knots. I - Introduction II - Definitions and condensed graphs II-1 Knots II-2 Sign of crossing points II-3 Chord diagrams II-4 Enlacement graphs II-5 Condensed graphs III - Realizability and construction III - 1 Realizability -
A Knot-Vice's Guide to Untangling Knot Theory, Undergraduate
A Knot-vice’s Guide to Untangling Knot Theory Rebecca Hardenbrook Department of Mathematics University of Utah Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 1 / 26 What is Not a Knot? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 2 / 26 What is a Knot? 2 A knot is an embedding of the circle in the Euclidean plane (R ). 3 Also defined as a closed, non-self-intersecting curve in R . 2 Represented by knot projections in R . Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 3 / 26 Why Knots? Late nineteenth century chemists and physicists believed that a substance known as aether existed throughout all of space. Could knots represent the elements? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 4 / 26 Why Knots? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 5 / 26 Why Knots? Unfortunately, no. Nevertheless, mathematicians continued to study knots! Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 6 / 26 Current Applications Natural knotting in DNA molecules (1980s). Credit: K. Kimura et al. (1999) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 7 / 26 Current Applications Chemical synthesis of knotted molecules – Dietrich-Buchecker and Sauvage (1988). Credit: J. Guo et al. (2010) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 8 / 26 Current Applications Use of lattice models, e.g. the Ising model (1925), and planar projection of knots to find a knot invariant via statistical mechanics. Credit: D. Chicherin, V.P. -
Gauss' Linking Number Revisited
October 18, 2011 9:17 WSPC/S0218-2165 134-JKTR S0218216511009261 Journal of Knot Theory and Its Ramifications Vol. 20, No. 10 (2011) 1325–1343 c World Scientific Publishing Company DOI: 10.1142/S0218216511009261 GAUSS’ LINKING NUMBER REVISITED RENZO L. RICCA∗ Department of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy [email protected] BERNARDO NIPOTI Department of Mathematics “F. Casorati”, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy Accepted 6 August 2010 ABSTRACT In this paper we provide a mathematical reconstruction of what might have been Gauss’ own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and inter- section number. The reconstruction presented here is entirely based on an accurate study of Gauss’ own work on terrestrial magnetism. A brief discussion of a possibly indepen- dent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided. Keywords: Linking number; potential; degree; signed crossings; intersection number; oriented area. Mathematics Subject Classification 2010: 57M25, 57M27, 78A25 1. Introduction The concept of linking number was introduced by Gauss in a brief note on his diary in 1833 (see Sec. 2 below), but no proof was given, neither of its derivation, nor of its topological meaning. Its derivation remained indeed a mystery. -
Knots: a Handout for Mathcircles
Knots: a handout for mathcircles Mladen Bestvina February 2003 1 Knots Informally, a knot is a knotted loop of string. You can create one easily enough in one of the following ways: • Take an extension cord, tie a knot in it, and then plug one end into the other. • Let your cat play with a ball of yarn for a while. Then find the two ends (good luck!) and tie them together. This is usually a very complicated knot. • Draw a diagram such as those pictured below. Such a diagram is a called a knot diagram or a knot projection. Trefoil and the figure 8 knot 1 The above two knots are the world's simplest knots. At the end of the handout you can see many more pictures of knots (from Robert Scharein's web site). The same picture contains many links as well. A link consists of several loops of string. Some links are so famous that they have names. For 2 2 3 example, 21 is the Hopf link, 51 is the Whitehead link, and 62 are the Bor- romean rings. They have the feature that individual strings (or components in mathematical parlance) are untangled (or unknotted) but you can't pull the strings apart without cutting. A bit of terminology: A crossing is a place where the knot crosses itself. The first number in knot's \name" is the number of crossings. Can you figure out the meaning of the other number(s)? 2 Reidemeister moves There are many knot diagrams representing the same knot. For example, both diagrams below represent the unknot. -
Computing the Writhing Number of a Polygonal Knot
Computing the Writhing Number of a Polygonal Knot ¡ ¡£¢ ¡ Pankaj K. Agarwal Herbert Edelsbrunner Yusu Wang Abstract Here the linking number, , is half the signed number of crossings between the two boundary curves of the ribbon, The writhing number measures the global geometry of a and the twisting number, , is half the average signed num- closed space curve or knot. We show that this measure is ber of local crossing between the two curves. The non-local related to the average winding number of its Gauss map. Us- crossings between the two curves correspond to crossings ing this relationship, we give an algorithm for computing the of the ribbon axis, which are counted by the writhing num- ¤ writhing number for a polygonal knot with edges in time ber, . A small subset of the mathematical literature on ¥§¦ ¨ roughly proportional to ¤ . We also implement a different, the subject can be found in [3, 20]. Besides the mathemat- simple algorithm and provide experimental evidence for its ical interest, the White Formula and the writhing number practical efficiency. have received attention both in physics and in biochemistry [17, 23, 26, 30]. For example, they are relevant in under- standing various geometric conformations we find for circu- 1 Introduction lar DNA in solution, as illustrated in Figure 1 taken from [7]. By representing DNA as a ribbon, the writhing number of its The writhing number is an attempt to capture the physical phenomenon that a cord tends to form loops and coils when it is twisted. We model the cord by a knot, which we define to be an oriented closed curve in three-dimensional space. -
A Remarkable 20-Crossing Tangle Shalom Eliahou, Jean Fromentin
A remarkable 20-crossing tangle Shalom Eliahou, Jean Fromentin To cite this version: Shalom Eliahou, Jean Fromentin. A remarkable 20-crossing tangle. 2016. hal-01382778v2 HAL Id: hal-01382778 https://hal.archives-ouvertes.fr/hal-01382778v2 Preprint submitted on 16 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A REMARKABLE 20-CROSSING TANGLE SHALOM ELIAHOU AND JEAN FROMENTIN Abstract. For any positive integer r, we exhibit a nontrivial knot Kr with r− r (20·2 1 +1) crossings whose Jones polynomial V (Kr) is equal to 1 modulo 2 . Our construction rests on a certain 20-crossing tangle T20 which is undetectable by the Kauffman bracket polynomial pair mod 2. 1. Introduction In [6], M. B. Thistlethwaite gave two 2–component links and one 3–component link which are nontrivial and yet have the same Jones polynomial as the corre- sponding unlink U 2 and U 3, respectively. These were the first known examples of nontrivial links undetectable by the Jones polynomial. Shortly thereafter, it was shown in [2] that, for any integer k ≥ 2, there exist infinitely many nontrivial k–component links whose Jones polynomial is equal to that of the k–component unlink U k. -
Introduction to Vassiliev Knot Invariants First Draft. Comments
Introduction to Vassiliev Knot Invariants First draft. Comments welcome. July 20, 2010 S. Chmutov S. Duzhin J. Mostovoy The Ohio State University, Mansfield Campus, 1680 Univer- sity Drive, Mansfield, OH 44906, USA E-mail address: [email protected] Steklov Institute of Mathematics, St. Petersburg Division, Fontanka 27, St. Petersburg, 191011, Russia E-mail address: [email protected] Departamento de Matematicas,´ CINVESTAV, Apartado Postal 14-740, C.P. 07000 Mexico,´ D.F. Mexico E-mail address: [email protected] Contents Preface 8 Part 1. Fundamentals Chapter 1. Knots and their relatives 15 1.1. Definitions and examples 15 § 1.2. Isotopy 16 § 1.3. Plane knot diagrams 19 § 1.4. Inverses and mirror images 21 § 1.5. Knot tables 23 § 1.6. Algebra of knots 25 § 1.7. Tangles, string links and braids 25 § 1.8. Variations 30 § Exercises 34 Chapter 2. Knot invariants 39 2.1. Definition and first examples 39 § 2.2. Linking number 40 § 2.3. Conway polynomial 43 § 2.4. Jones polynomial 45 § 2.5. Algebra of knot invariants 47 § 2.6. Quantum invariants 47 § 2.7. Two-variable link polynomials 55 § Exercises 62 3 4 Contents Chapter 3. Finite type invariants 69 3.1. Definition of Vassiliev invariants 69 § 3.2. Algebra of Vassiliev invariants 72 § 3.3. Vassiliev invariants of degrees 0, 1 and 2 76 § 3.4. Chord diagrams 78 § 3.5. Invariants of framed knots 80 § 3.6. Classical knot polynomials as Vassiliev invariants 82 § 3.7. Actuality tables 88 § 3.8. Vassiliev invariants of tangles 91 § Exercises 93 Chapter 4. -
1 Introduction
Contents | 1 1 Introduction Is the “cable spaghetti” on the floor really knotted or is it enough to pull on both ends of the wire to completely unfold it? Since its invention, knot theory has always been a place of interdisciplinarity. The first knot tables composed by Tait have been motivated by Lord Kelvin’s theory of atoms. But it was not until a century ago that tools for rigorously distinguishing the trefoil and its mirror image as members of two different knot classes were derived. In the first half of the twentieth century, knot theory seemed to be a place mainly drivenby algebraic and combinatorial arguments, mentioning the contributions of Alexander, Reidemeister, Seifert, Schubert, and many others. Besides the development of higher dimensional knot theory, the search for new knot invariants has been a major endeav- our since about 1960. At the same time, connections to applications in DNA biology and statistical physics have surfaced. DNA biology had made a huge progress and it was well un- derstood that the topology of DNA strands matters: modeling of the interplay between molecules and enzymes such as topoisomerases necessarily involves notions of ‘knot- tedness’. Any configuration involving long strands or flexible ropes with a relatively small diameter leads to a mathematical model in terms of curves. Therefore knots appear almost naturally in this context and require techniques from algebra, (differential) geometry, analysis, combinatorics, and computational mathematics. The discovery of the Jones polynomial in 1984 has led to the great popularity of knot theory not only amongst mathematicians and stimulated many activities in this direction. -
Invariants of Knots and 3-Manifolds: Survey on 3-Manifolds
Invariants of knots and 3-manifolds: Survey on 3-manifolds Wolfgang Lück Bonn Germany email [email protected] http://131.220.77.52/lueck/ Bonn, 10. & 12. April 2018 Wolfgang Lück (MI, Bonn) Survey on 3-manifolds Bonn, 10. & 12. April 2018 1 / 44 Tentative plan of the course title date lecturer Introduction to 3-manifolds I & II April, 10 & 12 Lück Cobordism theory and the April, 17 Lück s-cobordism theorem Introduction to Surgery theory April 19 Lück L2-Betti numbers April, 24 & 26 Lück Introduction to Knots and Links May 3 Teichner Knot invariants I May, 8 Teichner Knot invariants II May,15 Teichner Introduction to knot concordance I May, 17 Teichner Whitehead torsion and L2-torsion I May 29th Lück L2-signatures I June 5 Teichner tba June, 7 tba Wolfgang Lück (MI, Bonn) Survey on 3-manifolds Bonn, 10. & 12. April 2018 2 / 44 title date lecturer Whitehead torsion and L2-torsion II June, 12 Lück L2-invariants und 3-manifolds I June, 14 Lück L2-invariants und 3-manifolds II June, 19 Lück L2-signatures II June, 21 Teichner L2-signatures as knot concordance June, 26 & 28 Teichner invariants I & II tba July, 3 tba Further aspects of L2-invariants July 10 Lück tba July 12 Teichner Open problems in low-dimensional July 17 & 19 Teichner topology No talks on May 1, May 10, May 22, May 24, May 31, July 5. On demand there can be a discussion session at the end of the Thursday lecture. Wolfgang Lück (MI, Bonn) Survey on 3-manifolds Bonn, 10. -
How Can We Say 2 Knots Are Not the Same?
How can we say 2 knots are not the same? SHRUTHI SRIDHAR What’s a knot? A knot is a smooth embedding of the circle S1 in IR3. A link is a smooth embedding of the disjoint union of more than one circle Intuitively, it’s a string knotted up with ends joined up. We represent it on a plane using curves and ‘crossings’. The unknot A ‘figure-8’ knot A ‘wild’ knot (not a knot for us) Hopf Link Two knots or links are the same if they have an ambient isotopy between them. Representing a knot Knots are represented on the plane with strands and crossings where 2 strands cross. We call this picture a knot diagram. Knots can have more than one representation. Reidemeister moves Operations on knot diagrams that don’t change the knot or link Reidemeister moves Theorem: (Reidemeister 1926) Two knot diagrams are of the same knot if and only if one can be obtained from the other through a series of Reidemeister moves. Crossing Number The minimum number of crossings required to represent a knot or link is called its crossing number. Knots arranged by crossing number: Knot Invariants A knot/link invariant is a property of a knot/link that is independent of representation. Trivial Examples: • Crossing number • Knot Representations / ~ where 2 representations are equivalent via Reidemester moves Tricolorability We say a knot is tricolorable if the strands in any projection can be colored with 3 colors such that every crossing has 1 or 3 colors and or the coloring uses more than one color. -
The Linking Number and the Writhe of Uniform Random Walks And
The linking number and the writhe of uniform random walks and polygons in confined spaces E. Panagiotou ,∗ K. C. Millett ,† S. Lambropoulou ‡ November 11, 2018 Abstract Random walks and polygons are used to model polymers. In this pa- per we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configura- tions as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking num- ber of oriented uniform random walks or polygons of length n, in a convex 2 confined space, are of the form O(n ). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form O(√n). Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length. 1 Introduction A polymer melt may consist of ring polymers (closed chains), linear polymers (open chains), or a mixed collection of ring and linear polymers. Polymer chains are long flexible molecules that impose spatial constraints on each other because they cannot intersect (de Gennes 1979, Rubinstein and Colby 2003).