Knot Theory and Physics 1 Introduction
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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/. -
The Future of Fundamental Physics
The Future of Fundamental Physics Nima Arkani-Hamed Abstract: Fundamental physics began the twentieth century with the twin revolutions of relativity and quantum mechanics, and much of the second half of the century was devoted to the con- struction of a theoretical structure unifying these radical ideas. But this foundation has also led us to a number of paradoxes in our understanding of nature. Attempts to make sense of quantum mechanics and gravity at the smallest distance scales lead inexorably to the conclusion that space- Downloaded from http://direct.mit.edu/daed/article-pdf/141/3/53/1830482/daed_a_00161.pdf by guest on 23 September 2021 time is an approximate notion that must emerge from more primitive building blocks. Further- more, violent short-distance quantum fluctuations in the vacuum seem to make the existence of a macroscopic world wildly implausible, and yet we live comfortably in a huge universe. What, if anything, tames these fluctuations? Why is there a macroscopic universe? These are two of the central theoretical challenges of fundamental physics in the twenty-½rst century. In this essay, I describe the circle of ideas surrounding these questions, as well as some of the theoretical and experimental fronts on which they are being attacked. Ever since Newton realized that the same force of gravity pulling down on an apple is also responsible for keeping the moon orbiting the Earth, funda- mental physics has been driven by the program of uni½cation: the realization that seemingly disparate phenomena are in fact different aspects of the same underlying cause. By the mid-1800s, electricity and magnetism were seen as different aspects of elec- tromagnetism, and a seemingly unrelated phenom- enon–light–was understood to be the undulation of electric and magnetic ½elds. -
Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
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. -
A Symmetry Motivated Link Table
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 August 2018 doi:10.20944/preprints201808.0265.v1 Peer-reviewed version available at Symmetry 2018, 10, 604; doi:10.3390/sym10110604 Article A Symmetry Motivated Link Table Shawn Witte1, Michelle Flanner2 and Mariel Vazquez1,2 1 UC Davis Mathematics 2 UC Davis Microbiology and Molecular Genetics * Correspondence: [email protected] Abstract: Proper identification of oriented knots and 2-component links requires a precise link 1 nomenclature. Motivated by questions arising in DNA topology, this study aims to produce a 2 nomenclature unambiguous with respect to link symmetries. For knots, this involves distinguishing 3 a knot type from its mirror image. In the case of 2-component links, there are up to sixteen possible 4 symmetry types for each topology. The study revisits the methods previously used to disambiguate 5 chiral knots and extends them to oriented 2-component links with up to nine crossings. Monte Carlo 6 simulations are used to report on writhe, a geometric indicator of chirality. There are ninety-two 7 prime 2-component links with up to nine crossings. Guided by geometrical data, linking number and 8 the symmetry groups of 2-component links, a canonical link diagram for each link type is proposed. 9 2 2 2 2 2 2 All diagrams but six were unambiguously chosen (815, 95, 934, 935, 939, and 941). We include complete 10 tables for prime knots with up to ten crossings and prime links with up to nine crossings. We also 11 prove a result on the behavior of the writhe under local lattice moves. -
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. -
Introduction to General Relativity
INTRODUCTION TO GENERAL RELATIVITY Gerard 't Hooft Institute for Theoretical Physics Utrecht University and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail: [email protected] internet: http://www.phys.uu.nl/~thooft/ Version November 2010 1 Prologue General relativity is a beautiful scheme for describing the gravitational ¯eld and the equations it obeys. Nowadays this theory is often used as a prototype for other, more intricate constructions to describe forces between elementary particles or other branches of fundamental physics. This is why in an introduction to general relativity it is of importance to separate as clearly as possible the various ingredients that together give shape to this paradigm. After explaining the physical motivations we ¯rst introduce curved coordinates, then add to this the notion of an a±ne connection ¯eld and only as a later step add to that the metric ¯eld. One then sees clearly how space and time get more and more structure, until ¯nally all we have to do is deduce Einstein's ¯eld equations. These notes materialized when I was asked to present some lectures on General Rela- tivity. Small changes were made over the years. I decided to make them freely available on the web, via my home page. Some readers expressed their irritation over the fact that after 12 pages I switch notation: the i in the time components of vectors disappears, and the metric becomes the ¡ + + + metric. Why this \inconsistency" in the notation? There were two reasons for this. The transition is made where we proceed from special relativity to general relativity. -
"Eternal" Questions in the XX-Century Theoretical Physics V
Philosophical roots of the "eternal" questions in the XX-century theoretical physics V. Ihnatovych Department of Philosophy, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine e-mail: [email protected] Abstract The evolution of theoretical physics in the XX century differs significantly from that in XVII-XIX centuries. While continuous progress is observed for theoretical physics in XVII-XIX centuries, modern physics contains many questions that have not been resolved despite many decades of discussion. Based upon the analysis of works by the founders of the XX-century physics, the conclusion is made that the roots of the "eternal" questions by the XX-century theoretical physics lie in the philosophy used by its founders. The conclusion is made about the need to use the ideas of philosophy that guided C. Huygens, I. Newton, W. Thomson (Lord Kelvin), J. K. Maxwell, and the other great physicists of the XVII-XIX centuries, in all areas of theoretical physics. 1. Classical Physics The history of theoretical physics begins in 1687 with the work “Mathematical Principles of Natural Philosophy” by Isaac Newton. Even today, this work is an example of what a full and consistent outline of the physical theory should be. It contains everything necessary for such an outline – definition of basic concepts, the complete list of underlying laws, presentation of methods of theoretical research, rigorous proofs. In the eighteenth century, such great physicists and mathematicians as Euler, D'Alembert, Lagrange, Laplace and others developed mechanics, hydrodynamics, acoustics and nebular mechanics on the basis of the ideas of Newton's “Principles”. -
The Universe Unveiled Given by Prof Carlo Contaldi
Friends of Imperial Theoretical Physics We are delighted to announce that the first FITP event of 2015 will be a talk entitled The Universe Unveiled given by Prof Carlo Contaldi. The event is free and open to all but please register by visiting the Eventbrite website via http://tinyurl.com/fitptalk2015. Date: 29th April 2015 Venue: Lecture Theatre 1, Blackett Laboratory, Physics Department, ICL Time: 7-8pm followed by a reception in the level 8 Common room Speaker: Professor Carlo Contaldi The Universe Unveiled The past 25 years have seen our understanding of the Universe we live in being revolutionised by a series of stunning observational campaigns and theoretical advances. We now know the composition, age, geometry and evolutionary history of the Universe to an astonishing degree of precision. A surprising aspect of this journey of discovery is that it has revealed some profound conundrums that challenge the most basic tenets of fundamental physics. We still do not understand the nature of 95% of the matter and energy that seems to fill the Universe, we still do not know why or how the Universe came into being, and we have yet to build a consistent "theory of everything" that can describe the evolution of the Universe during the first few instances after the Big Bang. In this lecture I will review what we know about the Universe today and discuss the exciting experimental and theoretical advances happening in cosmology, including the controversy surrounding last year's BICEP2 "discovery". Biography of the speaker: Professor Contaldi gained his PhD in theoretical physics in 2000 at Imperial College working on theories describing the formation of structures in the universe. -
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. -
MUTATIONS of LINKS in GENUS 2 HANDLEBODIES 1. Introduction
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, January 1999, Pages 309{314 S 0002-9939(99)04871-6 MUTATIONS OF LINKS IN GENUS 2 HANDLEBODIES D. COOPER AND W. B. R. LICKORISH (Communicated by Ronald A. Fintushel) Abstract. A short proof is given to show that a link in the 3-sphere and any link related to it by genus 2 mutation have the same Alexander polynomial. This verifies a deduction from the solution to the Melvin-Morton conjecture. The proof here extends to show that the link signatures are likewise the same and that these results extend to links in a homology 3-sphere. 1. Introduction Suppose L is an oriented link in a genus 2 handlebody H that is contained, in some arbitrary (complicated) way, in S3.Letρbe the involution of H depicted abstractly in Figure 1 as a π-rotation about the axis shown. The pair of links L and ρL is said to be related by a genus 2 mutation. The first purpose of this note is to prove, by means of long established techniques of classical knot theory, that L and ρL always have the same Alexander polynomial. As described briefly below, this actual result for knots can also be deduced from the recent solution to a conjecture, of P. M. Melvin and H. R. Morton, that posed a problem in the realm of Vassiliev invariants. It is impressive that this simple result, readily expressible in the language of the classical knot theory that predates the Jones polynomial, should have emerged from the technicalities of Vassiliev invariants.