Classifying and Applying Rational Knots and Rational Tangles
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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. -
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. -
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring Of
CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring of Pretzel Knots A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mathematics By Robert Ostrander December 2013 The thesis of Robert Ostrander is approved: |||||||||||||||||| |||||||| Dr. Alberto Candel Date |||||||||||||||||| |||||||| Dr. Terry Fuller Date |||||||||||||||||| |||||||| Dr. Magnhild Lien, Chair Date California State University, Northridge ii Dedications I dedicate this thesis to my family and friends for all the help and support they have given me. iii Acknowledgments iv Table of Contents Signature Page ii Dedications iii Acknowledgements iv Abstract vi Introduction 1 1 Definitions and Background 2 1.1 Knots . .2 1.1.1 Composition of knots . .4 1.1.2 Links . .5 1.1.3 Torus Knots . .6 1.1.4 Reidemeister Moves . .7 2 Properties of Knots 9 2.0.5 Knot Invariants . .9 3 p-Coloring of Pretzel Knots 19 3.0.6 Pretzel Knots . 19 3.0.7 (p1, p2, p3) Pretzel Knots . 23 3.0.8 Applications of Theorem 6 . 30 3.0.9 (p1, p2, p3, p4) Pretzel Knots . 31 Appendix 49 v Abstract P coloring of Pretzel Knots by Robert Ostrander Master of Science in Mathematics In this thesis we give a brief introduction to knot theory. We define knot invariants and give examples of different types of knot invariants which can be used to distinguish knots. We look at colorability of knots and generalize this to p-colorability. We focus on 3-strand pretzel knots and apply techniques of linear algebra to prove theorems about p-colorability of these knots. -
Jhep07(2018)145
Published for SISSA by Springer Received: April 13, 2018 Accepted: July 16, 2018 Published: July 23, 2018 Symmetry enhancement and closing of knots in 3d/3d JHEP07(2018)145 correspondence Dongmin Ganga and Kazuya Yonekurab aCenter for Theorectical Physics, Seoul National University, Seoul 08826, Korea bKavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8583, Japan E-mail: [email protected], [email protected] Abstract: We revisit Dimofte-Gaiotto-Gukov's construction of 3d gauge theories asso- ciated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d N = (2; 0) theory of A1-type on a 3-manifold, we propose a topological criterion for SU(2)=SO(3) flavor symmetry enhancement for the u(1) symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the con- struction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising SU(3) symmetry en- hancement which is unexpected from the 6d viewpoint. Keywords: Chern-Simons Theories, Conformal Field Theory, Duality in Gauge Field Theories, M-Theory ArXiv ePrint: 1803.04009 Open Access, c The Authors. -
Matrix Integrals and Knot Theory 1
Matrix Integrals and Knot Theory 1 Matrix Integrals and Knot Theory Paul Zinn-Justin et Jean-Bernard Zuber math-ph/9904019 (Discr. Math. 2001) math-ph/0002020 (J.Kn.Th.Ramif. 2000) P. Z.-J. (math-ph/9910010, math-ph/0106005) J. L. Jacobsen & P. Z.-J. (math-ph/0102015, math-ph/0104009). math-ph/0303049 (J.Kn.Th.Ramif. 2004) G. Schaeffer and P. Z.-J., math-ph/0304034 P. Z.-J. and J.-B. Z., review article in preparation Carghjese, March 25, 2009 Matrix Integrals and Knot Theory 2 Courtesy of “U Rasaghiu” Matrix Integrals and Knot Theory 3 Main idea : Use combinatorial tools of Quantum Field Theory in Knot Theory Plan I Knot Theory : a few definitions II Matrix integrals and Link diagrams 1 2 4 dMeNtr( 2 M + gM ) N N matrices, N ∞ − × → RemovalsR of redundancies reproduces recent results of Sundberg & Thistlethwaite (1998) ⇒ based on Tutte (1963) III Virtual knots and links : counting and invariants Matrix Integrals and Knot Theory 4 Basics of Knot Theory a knot , links and tangles Equivalence up to isotopy Problem : Count topologically inequivalent knots, links and tangles Represent knots etc by their planar projection with minimal number of over/under-crossings Theorem Two projections represent the same knot/link iff they may be transformed into one another by a sequence of Reidemeister moves : ; ; Matrix Integrals and Knot Theory 5 Avoid redundancies by keeping only prime links (i.e. which cannot be factored) Consider the subclass of alternating knots, links and tangles, in which one meets alternatingly over- and under-crossings. For n 8 (resp. -
Knot Theory and the Alexander Polynomial
Knot Theory and the Alexander Polynomial Reagin Taylor McNeill Submitted to the Department of Mathematics of Smith College in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Elizabeth Denne, Faculty Advisor April 15, 2008 i Acknowledgments First and foremost I would like to thank Elizabeth Denne for her guidance through this project. Her endless help and high expectations brought this project to where it stands. I would Like to thank David Cohen for his support thoughout this project and through- out my mathematical career. His humor, skepticism and advice is surely worth the $.25 fee. I would also like to thank my professors, peers, housemates, and friends, particularly Kelsey Hattam and Katy Gerecht, for supporting me throughout the year, and especially for tolerating my temporary insanity during the final weeks of writing. Contents 1 Introduction 1 2 Defining Knots and Links 3 2.1 KnotDiagramsandKnotEquivalence . ... 3 2.2 Links, Orientation, and Connected Sum . ..... 8 3 Seifert Surfaces and Knot Genus 12 3.1 SeifertSurfaces ................................. 12 3.2 Surgery ...................................... 14 3.3 Knot Genus and Factorization . 16 3.4 Linkingnumber.................................. 17 3.5 Homology ..................................... 19 3.6 TheSeifertMatrix ................................ 21 3.7 TheAlexanderPolynomial. 27 4 Resolving Trees 31 4.1 Resolving Trees and the Conway Polynomial . ..... 31 4.2 TheAlexanderPolynomial. 34 5 Algebraic and Topological Tools 36 5.1 FreeGroupsandQuotients . 36 5.2 TheFundamentalGroup. .. .. .. .. .. .. .. .. 40 ii iii 6 Knot Groups 49 6.1 TwoPresentations ................................ 49 6.2 The Fundamental Group of the Knot Complement . 54 7 The Fox Calculus and Alexander Ideals 59 7.1 TheFreeCalculus ............................... -
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. -
An Introduction to the Theory of Knots
An Introduction to the Theory of Knots Giovanni De Santi December 11, 2002 Figure 1: Escher’s Knots, 1965 1 1 Knot Theory Knot theory is an appealing subject because the objects studied are familiar in everyday physical space. Although the subject matter of knot theory is familiar to everyone and its problems are easily stated, arising not only in many branches of mathematics but also in such diverse fields as biology, chemistry, and physics, it is often unclear how to apply mathematical techniques even to the most basic problems. We proceed to present these mathematical techniques. 1.1 Knots The intuitive notion of a knot is that of a knotted loop of rope. This notion leads naturally to the definition of a knot as a continuous simple closed curve in R3. Such a curve consists of a continuous function f : [0, 1] → R3 with f(0) = f(1) and with f(x) = f(y) implying one of three possibilities: 1. x = y 2. x = 0 and y = 1 3. x = 1 and y = 0 Unfortunately this definition admits pathological or so called wild knots into our studies. The remedies are either to introduce the concept of differentiability or to use polygonal curves instead of differentiable ones in the definition. The simplest definitions in knot theory are based on the latter approach. Definition 1.1 (knot) A knot is a simple closed polygonal curve in R3. The ordered set (p1, p2, . , pn) defines a knot; the knot being the union of the line segments [p1, p2], [p2, p3],..., [pn−1, pn], and [pn, p1]. -
Arxiv:2108.09698V1 [Math.GT] 22 Aug 2021
Submitted to Topology Proceedings THE TABULATION OF PRIME KNOT PROJECTIONS WITH THEIR MIRROR IMAGES UP TO EIGHT DOUBLE POINTS NOBORU ITO AND YUSUKE TAKIMURA Abstract. This paper provides the complete table of prime knot projections with their mirror images, without redundancy, up to eight double points systematically thorough a finite procedure by flypes. In this paper, we show how to tabulate the knot projec- tions up to eight double points by listing tangles with at most four double points by an approach with respect to rational tangles of J. H. Conway. In other words, for a given prime knot projection of an alternating knot, we show how to enumerate possible projec- tions of the alternating knot. Also to tabulate knot projections up to ambient isotopy, we introduce arrow diagrams (oriented Gauss diagrams) of knot projections having no over/under information of each crossing, which were originally introduced as arrow diagrams of knot diagrams by M. Polyak and O. Viro. Each arrow diagram of a knot projection completely detects the difference between the knot projection and its mirror image. 1. Introduction arXiv:2108.09698v1 [math.GT] 22 Aug 2021 Arnold ([2, Figure 53], [3, Figure 15]) obtained a table of reduced knot projections (equivalently, reduced generic immersed spherical curves) up to seven double points. In Arnold’s table, the number of prime knot pro- jections with seven double points is six. However, this table is incomplete (see Figure 1, which equals [2, Figure 53]). Key words and phrases. knot projection; tabulation; flype MSC 2010: 57M25, 57Q35. 1 2 NOBORU ITO AND YUSUKE TAKIMURA Figure 1. -
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 .