On Knots That Are Universal
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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/. -
Mutant Knots and Intersection Graphs 1 Introduction
Mutant knots and intersection graphs S. V. CHMUTOV S. K. LANDO We prove that if a finite order knot invariant does not distinguish mutant knots, then the corresponding weight system depends on the intersection graph of a chord diagram rather than on the diagram itself. Conversely, if we have a weight system depending only on the intersection graphs of chord diagrams, then the composition of such a weight system with the Kontsevich invariant determines a knot invariant that does not distinguish mutant knots. Thus, an equivalence between finite order invariants not distinguishing mutants and weight systems depending on intersections graphs only is established. We discuss relationship between our results and certain Lie algebra weight systems. 57M15; 57M25 1 Introduction Below, we use standard notions of the theory of finite order, or Vassiliev, invariants of knots in 3-space; their definitions can be found, for example, in [6] or [14], and we recall them briefly in Section 2. All knots are assumed to be oriented. Two knots are said to be mutant if they differ by a rotation of a tangle with four endpoints about either a vertical axis, or a horizontal axis, or an axis perpendicular to the paper. If necessary, the orientation inside the tangle may be replaced by the opposite one. Here is a famous example of mutant knots, the Conway (11n34) knot C of genus 3, and Kinoshita–Terasaka (11n42) knot KT of genus 2 (see [1]). C = KT = Note that the change of the orientation of a knot can be achieved by a mutation in the complement to a trivial tangle. -
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. -
Forming the Borromean Rings out of Polygonal Unknots
Forming the Borromean Rings out of polygonal unknots Hugh Howards 1 Introduction The Borromean Rings in Figure 1 appear to be made out of circles, but a result of Freedman and Skora shows that this is an optical illusion (see [F] or [H]). The Borromean Rings are a special type of Brunninan Link: a link of n components is one which is not an unlink, but for which every sublink of n − 1 components is an unlink. There are an infinite number of distinct Brunnian links of n components for n ≥ 3, but the Borromean Rings are the most famous example. This fact that the Borromean Rings cannot be formed from circles often comes as a surprise, but then we come to the contrasting result that although it cannot be built out of circles, the Borromean Rings can be built out of convex curves, for example, one can form it from two circles and an ellipse. Although it is only one out of an infinite number of Brunnian links of three Figure 1: The Borromean Rings. 1 components, it is the only one which can be built out of convex compo- nents [H]. The convexity result is, in fact a bit stronger and shows that no 4 component Brunnian link can be made out of convex components and Davis generalizes this result to 5 components in [D]. While this shows it is in some sense hard to form most Brunnian links out of certain shapes, the Borromean Rings leave some flexibility. This leads to the question of what shapes can be used to form the Borromean Rings and the following surprising conjecture of Matthew Cook at the California Institute of Technology. -
Hyperbolic Structures from Link Diagrams
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2012 Hyperbolic Structures from Link Diagrams Anastasiia Tsvietkova [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Geometry and Topology Commons Recommended Citation Tsvietkova, Anastasiia, "Hyperbolic Structures from Link Diagrams. " PhD diss., University of Tennessee, 2012. https://trace.tennessee.edu/utk_graddiss/1361 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Anastasiia Tsvietkova entitled "Hyperbolic Structures from Link Diagrams." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Mathematics. Morwen B. Thistlethwaite, Major Professor We have read this dissertation and recommend its acceptance: Conrad P. Plaut, James Conant, Michael Berry Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Hyperbolic Structures from Link Diagrams A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Anastasiia Tsvietkova May 2012 Copyright ©2012 by Anastasiia Tsvietkova. All rights reserved. ii Acknowledgements I am deeply thankful to Morwen Thistlethwaite, whose thoughtful guidance and generous advice made this research possible. -
Rock-Paper-Scissors Meets Borromean Rings
ROCK-PAPER-SCISSORS MEETS BORROMEAN RINGS MARC CHAMBERLAND AND EUGENE A. HERMAN 1. Introduction Directed graphs with an odd number of vertices n, where each vertex has both (n − 1)/2 incoming and outgoing edges, have a rich structure. We were lead to their study by both the Borromean rings and the game rock- paper-scissors. An interesting interplay between groups, graphs, topological links, and matrices reveals the structure of these objects, and for larger values of n, extensive computation produces some surprises. Perhaps most surprising is how few of the larger graphs have any symmetry and those with symmetry possess very little. In the final section, we dramatically sped up the computation by first computing a “profile” for each graph. 2. Three Weapons Let’s start with the two-player game rock-paper-scissors or RPS(3). The players simultaneously put their hands in one of three positions: rock (fist), paper (flat palm), or scissors (fist with the index and middle fingers sticking out). The winner of the game is decided as follows: paper covers rock, rock smashes scissors, and scissors cut paper. Mathematically, this game is referred to as a balanced tournament: with an odd number n of weapons, each weapon beats (n−1)/2 weapons and loses to the same number. This mutual dominance/submission connects RPS(3) with a seemingly disparate object: Borromean rings. Figure 1. Borromean Rings. 1 2 MARCCHAMBERLANDANDEUGENEA.HERMAN The Borromean rings in Figure 1 consist of three unknots in which the red ring lies on top of the blue ring, the blue on top of the green, and the green on top of the red. -
ON TANGLES and MATROIDS 1. Introduction Two Binary Operations on Matroids, the Two-Sum and the Tensor Product, Were Introduced I
ON TANGLES AND MATROIDS STEPHEN HUGGETT School of Mathematics and Statistics University of Plymouth, Plymouth PL4 8AA, Devon ABSTRACT Given matroids M and N there are two operations M ⊕2 N and M ⊗ N. When M and N are the cycle matroids of planar graphs these operations have interesting interpretations on the corresponding link diagrams. In fact, given a planar graph there are two well-established methods of generating an alternating link diagram, and in each case the Tutte polynomial of the graph is related to polynomial invariant (Jones or homfly) of the link. Switching from one of these methods to the other corresponds in knot theory to tangle insertion in the link diagrams, and in combinatorics to the tensor product of the cycle matroids of the graphs. Keywords: tangles, knots, Jones polynomials, Tutte polynomials, matroids 1. Introduction Two binary operations on matroids, the two-sum and the tensor product, were introduced in [13] and [4]. In the case when the matroids are the cycle matroids of graphs these give rise to corresponding operations on graphs. Strictly speaking, however, these operations on graphs are not in general well defined, being sensitive to the Whitney twist. Consider the two-sum for a moment: when it is well defined it can be seen simply in terms of the replacement of an edge by a new subgraph: a special case of this is the operation leading to the homeomorphic graphs studied in [11]. Planar graphs define alternating link diagrams, and so the two-sum and tensor product operations are defined on these too: they correspond to insertions of tangles. -
TANGLES for KNOTS and LINKS 1. Introduction It Is
TANGLES FOR KNOTS AND LINKS ZACHARY ABEL 1. Introduction It is often useful to discuss only small \pieces" of a link or a link diagram while disregarding everything else. For example, the Reidemeister moves describe manipulations surrounding at most 3 crossings, and the skein relations for the Jones and Conway polynomials discuss modifications on one crossing at a time. Tangles may be thought of as small pieces or a local pictures of knots or links, and they provide a useful language to describe the above manipulations. But the power of tangles in knot and link theory extends far beyond simple diagrammatic convenience, and this article provides a short survey of some of these applications. 2. Tangles As Used in Knot Theory Formally, we define a tangle as follows (in this article, everything is in the PL category): Definition 1. A tangle is a pair (A; t) where A =∼ B3 is a closed 3-ball and t is a proper 1-submanifold with @t 6= ;. Note that t may be disconnected and may contain closed loops in the interior of A. We consider two tangles (A; t) and (B; u) equivalent if they are isotopic as pairs of embedded manifolds. An n-string tangle consists of exactly n arcs and 2n boundary points (and no closed loops), and the trivial n-string 2 tangle is the tangle (B ; fp1; : : : ; png) × [0; 1] consisting of n parallel, unintertwined segments. See Figure 1 for examples of tangles. Note that with an appropriate isotopy any tangle (A; t) may be transformed into an equivalent tangle so that A is the unit ball in R3, @t = A \ t lies on the great circle in the xy-plane, and the projection (x; y; z) 7! (x; y) is a regular projection for t; thus, tangle diagrams as shown in Figure 1 are justified for all tangles. -
Knot and Link Tricolorability Danielle Brushaber Mckenzie Hennen Molly Petersen Faculty Mentor: Carolyn Otto University of Wisconsin-Eau Claire
Knot and Link Tricolorability Danielle Brushaber McKenzie Hennen Molly Petersen Faculty Mentor: Carolyn Otto University of Wisconsin-Eau Claire Problem & Importance Colorability Tables of Characteristics Theorem: For WH 51 with n twists, WH 5 is tricolorable when Knot Theory, a field of Topology, can be used to model Original Knot The unknot is not tricolorable, therefore anything that is tri- 1 colorable cannot be the unknot. The prime factors of the and understand how enzymes (called topoisomerases) work n = 3k + 1 where k ∈ N ∪ {0}. in DNA processes to untangle or repair strands of DNA. In a determinant of the knot or link provides the colorability. For human cell nucleus, the DNA is linear, so the knots can slip off example, if a knot’s determinant is 21, it is 3-colorable (tricol- orable), and 7-colorable. This is known by the theorem that Proof the end, and it is difficult to recognize what the enzymes do. Consider WH 5 where n is the number of the determinant of a knot is 0 mod n if and only if the knot 1 However, the DNA in mitochon- full positive twists. n is n-colorable. dria is circular, along with prokary- Link Colorability Det(L) Unknot/Link (n = 1...n = k) otic cells (bacteria), so the enzyme L Number Theorem: If det(L) = 0, then L is WOLOG, let WH 51 be colored in this way, processes are more noticeable in 3 3 3 1 n 1 n-colorable for all n. excluding coloring the twist component. knots in this type of DNA. -
Character Varieties of Knots and Links with Symmetries Leona H
Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2017 Character Varieties of Knots and Links with Symmetries Leona H. Sparaco Follow this and additional works at the DigiNole: FSU's Digital Repository. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES CHARACTER VARIETIES OF KNOTS AND LINKS WITH SYMMETRIES By LEONA SPARACO A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2017 Copyright c 2017 Leona Sparaco. All Rights Reserved. Leona Sparaco defended this dissertation on July 6, 2017. The members of the supervisory committee were: Kathleen Petersen Professor Directing Dissertation Kristine Harper University Representative Sam Ballas Committee Member Philip Bowers Committee Member Eriko Hironaka Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii ACKNOWLEDGMENTS I would not have been able to write this paper without the help and support of many people. I would first like to thank my major professor, Dr. Kathleen Petersen. I fell in love with topology when I was in her topology class. I would also like to thank the members of my committee and Dr. Case for their support over the years. I would like to thank my family for their love and support. Finally I want to thank all the wonderful friends I have made here while at Florida State University. iii TABLE OF CONTENTS List of Figures . v Abstract . -
Using Link Invariants to Determine Shapes for Links
USING LINK INVARIANTS TO DETERMINE SHAPES FOR LINKS DAN TATING Abstract. In the tables of two component links up to nine cross- ing there are 92 prime links. These different links take a variety of forms and, inspired by a proof that Borromean circles are im- possible, the questions are raised: Is there a possibility for the components of links to be geometric shapes? How can we deter- mine if a link can be formed by a shape? Is there a link invariant we can use for this determination? These questions are answered with proofs along with a tabulation of the link invariants; Conway polynomial, linking number, and enhanced linking number, in the following report on “Using Link Invariants to Determine Shapes for Links”. 1. An Introduction to Links By definition, a link is a set of knotted loops all tangled together. Two links are equivalent if we can deform the one link to the other link without ever having any one of the loops intersect itself or any of the other loops in the process [1].We tabulate links by using projections that minimize the number of crossings. Some basic links are shown below. Notice how each of these links has two loops, or components. Al- though links can have any finite number of components, we will focus This research was conducted as part of a 2003 REU at CSU, Chico supported by the MAA’s program for Strengthening Underrepresented Minority Mathematics Achievement and with funding from the NSF and NSA. 1 2 DAN TATING on links of two and three components. -
Visualization of Seifert Surfaces Jarke J
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 1, NO. X, AUGUST 2006 1 Visualization of Seifert Surfaces Jarke J. van Wijk Arjeh M. Cohen Technische Universiteit Eindhoven Abstract— The genus of a knot or link can be defined via Oriented surfaces whose boundaries are a knot K are called Seifert surfaces. A Seifert surface of a knot or link is an Seifert surfaces of K, after Herbert Seifert, who gave an algo- oriented surface whose boundary coincides with that knot or rithm to construct such a surface from a diagram describing the link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understand knot in 1934 [13]. His algorithm is easy to understand, but this their shape and structure. In this article the visualization of does not hold for the geometric shape of the resulting surfaces. such surfaces is discussed. A method is presented to produce Texts on knot theory only contain schematic drawings, from different styles of surface for knots and links, starting from the which it is hard to capture what is going on. In the cited paper, so-called braid representation. Application of Seifert's algorithm Seifert also introduced the notion of the genus of a knot as leads to depictions that show the structure of the knot and the surface, while successive relaxation via a physically based the minimal genus of a Seifert surface. The present article is model gives shapes that are natural and resemble the familiar dedicated to the visualization of Seifert surfaces, as well as representations of knots.