Knots Notes by Richard Hepworth
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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/. -
Oriented Pair (S 3,S1); Two Knots Are Regarded As
S-EQUIVALENCE OF KNOTS C. KEARTON Abstract. S-equivalence of classical knots is investigated, as well as its rela- tionship with mutation and the unknotting number. Furthermore, we identify the kernel of Bredon’s double suspension map, and give a geometric relation between slice and algebraically slice knots. Finally, we show that every knot is S-equivalent to a prime knot. 1. Introduction An oriented knot k is a smooth (or PL) oriented pair S3,S1; two knots are regarded as the same if there is an orientation preserving diffeomorphism sending one onto the other. An unoriented knot k is defined in the same way, but without regard to the orientation of S1. Every oriented knot is spanned by an oriented surface, a Seifert surface, and this gives rise to a matrix of linking numbers called a Seifert matrix. Any two Seifert matrices of the same knot are S-equivalent: the definition of S-equivalence is given in, for example, [14, 21, 11]. It is the equivalence relation generated by ambient surgery on a Seifert surface of the knot. In [19], two oriented knots are defined to be S-equivalent if their Seifert matrices are S- equivalent, and the following result is proved. Theorem 1. Two oriented knots are S-equivalent if and only if they are related by a sequence of doubled-delta moves shown in Figure 1. .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .. .... .... .... .... .... .... .... .... .... ... -
Arxiv:1006.4176V4 [Math.GT] 4 Nov 2011 Nepnigo Uho Ntter.Freape .W Alexander Mo W
UNKNOTTING UNKNOTS A. HENRICH AND L. KAUFFMAN Abstract. A knot is an an embedding of a circle into three–dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surprisingly difficult, since it has been shown that knot diagrams may need to be made more complicated before they may be simplified. We do not yet know, however, how much more complicated they must get. We give an introduction to the work of Dynnikov who discovered the key use of arc–presentations to solve the problem of finding a way to de- tect the unknot directly from a diagram of the knot. Using Dynnikov’s work, we show how to obtain a quadratic upper bound for the number of crossings that must be introduced into a sequence of unknotting moves. We also apply Dynnikov’s results to find an upper bound for the number of moves required in an unknotting sequence. 1. Introduction When one first delves into the theory of knots, one learns that knots are typically studied using their diagrams. The first question that arises when considering these knot diagrams is: how can we tell if two knot diagrams represent the same knot? Fortunately, we have a partial answer to this question. Two knot diagrams represent the same knot in R3 if and only if they can be related by the Reidemeister moves, pictured below. -
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. -
Deep Learning the Hyperbolic Volume of a Knot Arxiv:1902.05547V3 [Hep-Th] 16 Sep 2019
Deep Learning the Hyperbolic Volume of a Knot Vishnu Jejjalaa;b , Arjun Karb , Onkar Parrikarb aMandelstam Institute for Theoretical Physics, School of Physics, NITheP, and CoE-MaSS, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa bDavid Rittenhouse Laboratory, University of Pennsylvania, 209 S 33rd Street, Philadelphia, PA 19104, USA E-mail: [email protected], [email protected], [email protected] Abstract: An important conjecture in knot theory relates the large-N, double scaling limit of the colored Jones polynomial JK;N (q) of a knot K to the hyperbolic volume of the knot complement, Vol(K). A less studied question is whether Vol(K) can be recovered directly from the original Jones polynomial (N = 2). In this report we use a deep neural network to approximate Vol(K) from the Jones polynomial. Our network is robust and correctly predicts the volume with 97:6% accuracy when training on 10% of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial. arXiv:1902.05547v3 [hep-th] 16 Sep 2019 Contents 1 Introduction1 2 Setup and Result3 3 Discussion7 A Overview of knot invariants9 B Neural networks 10 B.1 Details of the network 12 C Other experiments 14 1 Introduction Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of these patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. In this work, we apply this idea to invariants in knot theory. -
Ribbon Concordances and Doubly Slice Knots
Ribbon concordances and doubly slice knots Ruppik, Benjamin Matthias Advisors: Arunima Ray & Peter Teichner The knot concordance group Glossary Definition 1. A knot K ⊂ S3 is called (topologically/smoothly) slice if it arises as the equatorial slice of a (locally flat/smooth) embedding of a 2-sphere in S4. – Abelian invariants: Algebraic invari- Proposition. Under the equivalence relation “K ∼ J iff K#(−J) is slice”, the commutative monoid ants extracted from the Alexander module Z 3 −1 A (K) = H1( \ K; [t, t ]) (i.e. from the n isotopy classes of o connected sum S Z K := 1 3 , # := oriented knots S ,→S of knots homology of the universal abelian cover of the becomes a group C := K/ ∼, the knot concordance group. knot complement). The neutral element is given by the (equivalence class) of the unknot, and the inverse of [K] is [rK], i.e. the – Blanchfield linking form: Sesquilin- mirror image of K with opposite orientation. ear form on the Alexander module ( ) sm top Bl: AZ(K) × AZ(K) → Q Z , the definition Remark. We should be careful and distinguish the smooth C and topologically locally flat C version, Z[Z] because there is a huge difference between those: Ker(Csm → Ctop) is infinitely generated! uses that AZ(K) is a Z[Z]-torsion module. – Levine-Tristram-signature: For ω ∈ S1, Some classical structure results: take the signature of the Hermitian matrix sm top ∼ ∞ ∞ ∞ T – There are surjective homomorphisms C → C → AC, where AC = Z ⊕ Z2 ⊕ Z4 is Levine’s algebraic (1 − ω)V + (1 − ω)V , where V is a Seifert concordance group. -
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. -
Tait's Flyping Conjecture for 4-Regular Graphs
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Journal of Combinatorial Theory, Series B 95 (2005) 318–332 www.elsevier.com/locate/jctb Tait’s flyping conjecture for 4-regular graphs Jörg Sawollek Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany Received 8 July 1998 Available online 19 July 2005 Abstract Tait’s flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S3. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are identical on the class of diagrams considered. © 2005 Elsevier Inc. All rights reserved. Keywords: Knotted graph; Alternating diagram; Flyping conjecture 0. Introduction Very early in the history of knot theory attention has been paid to alternating diagrams of knots and links. At the end of the 19th century Tait [21] stated several famous conjectures on alternating link diagrams that could not be verified for about a century. The conjectures concerning minimal crossing numbers of reduced, alternating link diagrams [15, Theorems A, B] have been proved independently by Thistlethwaite [22], Murasugi [15], and Kauffman [6]. Tait’s flyping conjecture, claiming that two reduced, alternating, prime diagrams of a given link can be connected by a finite sequence of so-called flypes (see [4, p. 311] for Tait’s original terminology), has been shown by Menasco and Thistlethwaite [14], and for a special case, namely, for well-connected diagrams, also by Schrijver [20]. -
On the Number of Unknot Diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar
On the number of unknot diagrams Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar To cite this version: Carolina Medina, Jorge Luis Ramírez Alfonsín, Gelasio Salazar. On the number of unknot diagrams. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2019, 33 (1), pp.306-326. 10.1137/17M115462X. hal-02049077 HAL Id: hal-02049077 https://hal.archives-ouvertes.fr/hal-02049077 Submitted on 26 Feb 2019 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. On the number of unknot diagrams Carolina Medina1, Jorge L. Ramírez-Alfonsín2,3, and Gelasio Salazar1,3 1Instituto de Física, UASLP. San Luis Potosí, Mexico, 78000. 2Institut Montpelliérain Alexander Grothendieck, Université de Montpellier. Place Eugèene Bataillon, 34095 Montpellier, France. 3Unité Mixte Internationale CNRS-CONACYT-UNAM “Laboratoire Solomon Lefschetz”. Cuernavaca, Mexico. October 17, 2017 Abstract Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2n diagrams. A folklore argument shows that at least one of these 2n diagrams is unknot, from which it follows that every diagram has finite unknotting number. -
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. -
An Introduction to Knot Theory and the Knot Group
AN INTRODUCTION TO KNOT THEORY AND THE KNOT GROUP LARSEN LINOV Abstract. This paper for the University of Chicago Math REU is an expos- itory introduction to knot theory. In the first section, definitions are given for knots and for fundamental concepts and examples in knot theory, and motivation is given for the second section. The second section applies the fun- damental group from algebraic topology to knots as a means to approach the basic problem of knot theory, and several important examples are given as well as a general method of computation for knot diagrams. This paper assumes knowledge in basic algebraic and general topology as well as group theory. Contents 1. Knots and Links 1 1.1. Examples of Knots 2 1.2. Links 3 1.3. Knot Invariants 4 2. Knot Groups and the Wirtinger Presentation 5 2.1. Preliminary Examples 5 2.2. The Wirtinger Presentation 6 2.3. Knot Groups for Torus Knots 9 Acknowledgements 10 References 10 1. Knots and Links We open with a definition: Definition 1.1. A knot is an embedding of the circle S1 in R3. The intuitive meaning behind a knot can be directly discerned from its name, as can the motivation for the concept. A mathematical knot is just like a knot of string in the real world, except that it has no thickness, is fixed in space, and most importantly forms a closed loop, without any loose ends. For mathematical con- venience, R3 in the definition is often replaced with its one-point compactification S3. Of course, knots in the real world are not fixed in space, and there is no interesting difference between, say, two knots that differ only by a translation.