DOCSLIB.ORG

Knots and (A Little Of) Braids

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Knots and (a little of) Braids Camila A. Ramirez University of Iowa February 28, 2016 C. Ramirez (UI) Knots and Braids February 28, 2016 1 / 38 Questions Given a tangled closed piece of string, is it really knotted or can it be untangled without having to cut it? Given two knotted pieces of closed string, when can we change one into the other? Does there exist an algorithm that can answer these questions? C. Ramirez (UI) Knots and Braids February 28, 2016 2 / 38 1 Introduction History 2 Knots Definition Projections Reidemeister Moves Knot Invariants Links More Knots 3 Braids Definition Braid Equivalence Braid Multiplication Closure of Braids 4 Applications C. Ramirez (UI) Knots and Braids February 28, 2016 3 / 38 Introduction History History In the late 19th century, physicists postulated that a substance called ether permeated all throughout space. In the 1860s, Lord Kelvin proposed that elements should have unique signatures based on how the atoms knotted up the ether around them. This theory led Scottish mathematical-physicist Peter Guthrie Tait to the creation of the first knot tables. In 1887 the Michelson-Morley experiment provided evidence against the existence of the ether, but it was too late for mathematicians, they were already tangled up! C. Ramirez (UI) Knots and Braids February 28, 2016 4 / 38 Knots Definition What is a knot? Take a piece of string and tie a knot on it. Glue the ends together. You have a "mathematical" knot! 3 Definition (A tame knot is a simple closed polygonal curve in R ) A knot is the union of the segments [p1; p2]; [p2; p3]; :::; [pn − 1; pn] of an ordered set of distinct points fp1; p2; :::; p3g in which each segment intersect exactly two others. C. Ramirez (UI) Knots and Braids February 28, 2016 5 / 38 Knots Projections Knot Projection Definition A knot projection is called a regular projection if no three points on the knot project to the same point and no vertex projects to the same point as any other point on the knot. Projections of a knot to the plane allow the representation of a knot as a knot diagram. C. Ramirez (UI) Knots and Braids February 28, 2016 6 / 38 Knots Projections Types of Knots Definition An oriented knot consists of a knot and an ordering of its vertices. Two orderings are called equivalent if they differ by a cyclic permutation. Definition An alternating knot is a knot with a diagram that has crossings alternating between over and under, for a fixed orientation. C. Ramirez (UI) Knots and Braids February 28, 2016 7 / 38 Knots Projections Knot Symmetry Definition A chiral knot is a knot that is not equivalent to its mirror image. An oriented knot that is equivalent to its mirror image is amphicheiral. C. Ramirez (UI) Knots and Braids February 28, 2016 8 / 38 Knots Projections Knot Addition: K = K1#K2 Definition A composite knot is a knot which can be formed by the connected sum of two or more nontrivial knots. The knots that make up a composite knot are called factor knots. The connected sum of two knots, K1 and K2 is formed by removing a small arc from each knot and then connecting the four endpoints by two new arcs in such way that no new crossings are introduced. C. Ramirez (UI) Knots and Braids February 28, 2016 9 / 38 Knots Projections Prime knots Definition A knot is called prime if for any decomposition as a connected sum, one of the factors is the unknot. Theorem (Prime Decomposition Theorem) Every knot can be decomposed as the connected sum of nontrivial prime knots. If K = K1#K2#:::#Kn and K = J1#:::J2#:::#Jm with each Ki and Ji nontrivial prime knots, then m = n and, after reordering each Ki is equivalent to Ji . C. Ramirez (UI) Knots and Braids February 28, 2016 10 / 38 Knots Projections Are two knots the same? Given a knot, is it the unknot? C. Ramirez (UI) Knots and Braids February 28, 2016 11 / 38 Knots Projections Equivalence of knots Definition Two knots K and J are called equivalent if there is a sequence of knots K = K0; K1; :::; Kn = J with each Ki + 1 an elementary deformation of Ki . Theorem If two knots K and J have identical diagrams, then they are equivalent. C. Ramirez (UI) Knots and Braids February 28, 2016 12 / 38 Knots Reidemeister Moves Reidemeister Moves Kurt Reidemeister (1893-1971) Reidemeister was a German mathematician, whose interests were mainly in combinatorial group theory, combinatorial topology, geometric group theory, and the foundations of geometry. C. Ramirez (UI) Knots and Braids February 28, 2016 13 / 38 Knots Reidemeister Moves Reidemeister Moves In 1926, Reidemeister (1927, J. W. Alexander and G. B. Briggs) demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves. Theorem Two knots (or links) are equivalent if and only if their diagrams are related by a sequence of Reidemeister moves. C. Ramirez (UI) Knots and Braids February 28, 2016 14 / 38 Knots Reidemeister Moves Example C. Ramirez (UI) Knots and Braids February 28, 2016 15 / 38 Knots Reidemeister Moves Example? Can you find a sequence of Reidemeister Moves deforming the right-handed trefoil into the left-handed trefoil? C. Ramirez (UI) Knots and Braids February 28, 2016 16 / 38 Knots Knot Invariants Ways of determining knot equivalence Definition Suppose that to each knot, K, we assign a specific quantity q(K). If for two equivalent knots the assigned quantities are always equal, then we call such a quantity a knot invariant. crossing number chirality unknotting number knot genus linking number Alexander polynomial tricolorability Jones polynomial stick number Homfly polynomial Question Does there exist a knot polynomial which distinguishes all knots from each other, or even simpler which distinguishes the unknot from all other knots? C. Ramirez (UI) Knots and Braids February 28, 2016 17 / 38 Knots Knot Invariants Crossing number The crossing number of a knot K, denoted c(K), is the least number of crossings that occur, ranging over all possible diagrams of a knot. Question: Is c(Ki #K2) = c(K1) + c(K2) for a composite knot? True for alternating knots (Kauffman, 1988) C. Ramirez (UI) Knots and Braids February 28, 2016 18 / 38 Knots Knot Invariants Unknotting number The unkotting number of a knot K, denoted u(K), is the least number of crossing changes that are required for the knot to become unknotted, ranging over all possible diagrams. The unknotting number of a knot is always less than half of its crossing number. Any composite knot has unknotting number at least two, and therefore every knot with unknotting number one is a prime knot. (Scharlemann 1985) u(K) is not necessarily realized in a projection with minimal crossing number. (S. Bleiler, Y. Nalsanishi 1983) C. Ramirez (UI) Knots and Braids February 28, 2016 19 / 38 Knots Knot Invariants Tricolorability A knot or link is called tricolorable if it is possible to do the following: 1 Color every strand in the diagram of the knot. 2 Use a total of three colors. 3 At each crossing, either make all the strands one color or make each strand a different color. Can you tricolor the figure eight knot? Theorem If a diagram of a knot K is tricorolable, then every diagram of K is tricolorable. C. Ramirez (UI) Knots and Braids February 28, 2016 20 / 38 Knots Links Links Definition A link is a collection of knots; the individual knots which make up a link are called the components of the link. C. Ramirez (UI) Knots and Braids February 28, 2016 21 / 38 Knots Links Linking number Given an oriented link we calculated its linking number in order to measure how linked up the components are. In order to do this, we first assign signs to the crossings as follows: Definition The linking number is a link invariant defined for a two-component oriented link as the sum of +1 crossings and -1 crossing over all crossings between the two links divided by 2. C. Ramirez (UI) Knots and Braids February 28, 2016 22 / 38 Knots Links Linking number C. Ramirez (UI) Knots and Braids February 28, 2016 23 / 38 Knots More Knots Other Types of Knots Stick Knot A stick knot is a knot formed out of straight sticks instead of flexible rope. The stick number of any knot is the least number of straight sticks necessary to form the knot. Example: The trefoil knot has a stick number of 6. Fact: No nontrivial knot can be formed by less than 6 sticks. C. Ramirez (UI) Knots and Braids February 28, 2016 24 / 38 Knots More Knots Other Types of Knots Wild Knots A wild knot is any knot that is not tame, meaning it cannot be 3 represented as a polygonal path in R . C. Ramirez (UI) Knots and Braids February 28, 2016 25 / 38 Braids Braids The idea behind braid theory is that braids can be organized into algebraic groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second braid on the twisted strings'. Even though braids are closely related to knots, the theory has developed semi-independently of knot theory. Braid groups were introduced explicitly by Emil Artin in 1925, although they were already implicit in Adolf Hurwitz's work as the fundamental group of a configuration space (1891). C. Ramirez (UI) Knots and Braids February 28, 2016 26 / 38 Braids Definition What is a braid? Take a unit cube, and place in it n strands of string, subject to the following conditions: 1 No part of any strand lies outside the cube.
Recommended publications
  • Conservation of Complex Knotting and Slipknotting Patterns in Proteins

    Conservation of Complex Knotting and Slipknotting Patterns in Proteins

    Conservation of complex knotting and PNAS PLUS slipknotting patterns in proteins Joanna I. Sułkowskaa,1,2, Eric J. Rawdonb,1, Kenneth C. Millettc, Jose N. Onuchicd,2, and Andrzej Stasiake aCenter for Theoretical Biological Physics, University of California at San Diego, La Jolla, CA 92037; bDepartment of Mathematics, University of St. Thomas, Saint Paul, MN 55105; cDepartment of Mathematics, University of California, Santa Barbara, CA 93106; dCenter for Theoretical Biological Physics , Rice University, Houston, TX 77005; and eCenter for Integrative Genomics, Faculty of Biology and Medicine, University of Lausanne, 1015 Lausanne, Switzerland Edited by* Michael S. Waterman, University of Southern California, Los Angeles, CA, and approved May 4, 2012 (received for review April 17, 2012) While analyzing all available protein structures for the presence of ers fixed configurations, in this case proteins in their native folded knots and slipknots, we detected a strict conservation of complex structures. Then they may be treated as frozen and thus unable to knotting patterns within and between several protein families undergo any deformation. despite their large sequence divergence. Because protein folding Several papers have described various interesting closure pathways leading to knotted native protein structures are slower procedures to capture the knot type of the native structure of and less efficient than those leading to unknotted proteins with a protein or a subchain of a closed chain (1, 3, 29–33). In general, similar size and sequence, the strict conservation of the knotting the strategy is to ensure that the closure procedure does not affect patterns indicates an important physiological role of knots and slip- the inherent entanglement in the analyzed protein chain or sub- knots in these proteins.
  • The Khovanov Homology of Rational Tangles

    The Khovanov Homology of Rational Tangles

    The Khovanov Homology of Rational Tangles Benjamin Thompson A thesis submitted for the degree of Bachelor of Philosophy with Honours in Pure Mathematics of The Australian National University October, 2016 Dedicated to my family. Even though they’ll never read it. “To feel fulfilled, you must first have a goal that needs fulfilling.” Hidetaka Miyazaki, Edge (280) “Sleep is good. And books are better.” (Tyrion) George R. R. Martin, A Clash of Kings “Let’s love ourselves then we can’t fail to make a better situation.” Lauryn Hill, Everything is Everything iv Declaration Except where otherwise stated, this thesis is my own work prepared under the supervision of Scott Morrison. Benjamin Thompson October, 2016 v vi Acknowledgements What a ride. Above all, I would like to thank my supervisor, Scott Morrison. This thesis would not have been written without your unflagging support, sublime feedback and sage advice. My thesis would have likely consisted only of uninspired exposition had you not provided a plethora of interesting potential topics at the start, and its overall polish would have likely diminished had you not kept me on track right to the end. You went above and beyond what I expected from a supervisor, and as a result I’ve had the busiest, but also best, year of my life so far. I must also extend a huge thanks to Tony Licata for working with me throughout the year too; hopefully we can figure out what’s really going on with the bigradings! So many people to thank, so little time. I thank Joan Licata for agreeing to run a Knot Theory course all those years ago.
  • Complete Invariant Graphs of Alternating Knots

    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, 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.
  • CROSSING CHANGES and MINIMAL DIAGRAMS the Unknotting

    CROSSING CHANGES and MINIMAL DIAGRAMS the Unknotting

    CROSSING CHANGES AND MINIMAL DIAGRAMS ISABEL K. DARCY The unknotting number of a knot is the minimal number of crossing changes needed to convert the knot into the unknot where the minimum is taken over all possible diagrams of the knot. For example, the minimal diagram of the knot 108 requires 3 crossing changes in order to change it to the unknot. Thus, by looking only at the minimal diagram of 108, it is clear that u(108) ≤ 3 (figure 1). Nakanishi [5] and Bleiler [2] proved that u(108) = 2. They found a non-minimal diagram of the knot 108 in which two crossing changes suffice to obtain the unknot (figure 2). Lower bounds on unknotting number can be found by looking at how knot invariants are affected by crossing changes. Nakanishi [5] and Bleiler [2] used signature to prove that u(108)=2. Figure 1. Minimal dia- Figure 2. A non-minimal dia- gram of knot 108 gram of knot 108 Bernhard [1] noticed that one can determine that u(108) = 2 by using a sequence of crossing changes within minimal diagrams and ambient isotopies as shown in figure. A crossing change within the minimal diagram of 108 results in the knot 62. We then use an ambient isotopy to change this non-minimal diagram of 62 to a minimal diagram. The unknot can then be obtained by changing one crossing within the minimal diagram of 62. Bernhard hypothesized that the unknotting number of a knot could be determined by only looking at minimal diagrams by using ambient isotopies between crossing changes.
  • A Symmetry Motivated Link Table

    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.
  • Knot Theory: an Introduction

    Knot Theory: an Introduction

    Knot theory: an introduction Zhen Huan Center for Mathematical Sciences Huazhong University of Science and Technology USTC, December 25, 2019 Knots in daily life Shoelace Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 2 / 40 Knots in daily life Braids Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 3 / 40 Knots in daily life Knot bread Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 4 / 40 Knots in daily life German bread: the pretzel Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 5 / 40 Knots in daily life Rope Mat Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 6 / 40 Knots in daily life Chinese Knots Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 7 / 40 Knots in daily life Knot bracelet Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 8 / 40 Knots in daily life Knitting Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 9 / 40 Knots in daily life More Knitting Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 10 / 40 Knots in daily life DNA Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 11 / 40 Knots in daily life Wire Mess Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 12 / 40 History of Knots Chinese talking knots (knotted strings) Inca Quipu Zhen Huan (HUST) Knot theory: an introduction USTC, December 25, 2019 13 / 40 History of Knots Endless Knot in Buddhism Zhen Huan (HUST) Knot theory: an introduction
  • Knots, Links, Spatial Graphs, and Algebraic Invariants

    Knots, Links, Spatial Graphs, and Algebraic Invariants

    689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society 689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 05C10, 57M15, 57M25, 57M27. Library of Congress Cataloging-in-Publication Data Names: Flapan, Erica, 1956- editor. Title: Knots, links, spatial graphs, and algebraic invariants : AMS special session on algebraic and combinatorial structures in knot theory, October 24-25, 2015, California State University, Fullerton, CA : AMS special session on spatial graphs, October 24-25, 2015, California State University, Fullerton, CA / Erica Flapan [and three others], editors. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Con- temporary mathematics ; volume 689 | Includes bibliographical references. Identifiers: LCCN 2016042011 | ISBN 9781470428471 (alk. paper) Subjects: LCSH: Knot theory–Congresses. | Link theory–Congresses. | Graph theory–Congresses. | Invariants–Congresses. | AMS: Combinatorics – Graph theory – Planar graphs; geometric and topological aspects of graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Relations with graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Knots and links in S3.msc| Manifolds and cell complexes – Low-dimensional topology – Invariants of knots and 3-manifolds.
  • Planar and Spherical Stick Indices of Knots

    Planar and Spherical Stick Indices of Knots

    PLANAR AND SPHERICAL STICK INDICES OF KNOTS COLIN ADAMS, DAN COLLINS, KATHERINE HAWKINS, CHARMAINE SIA, ROB SILVERSMITH, AND BENA TSHISHIKU Abstract. The stick index of a knot is the least number of line segments required to build the knot in space. We define two analogous 2-dimensional invariants, the planar stick index, which is the least number of line segments in the plane to build a projection, and the spherical stick index, which is the least number of great circle arcs to build a projection on the sphere. We find bounds on these quantities in terms of other knot invariants, and give planar stick and spherical stick constructions for torus knots and for compositions of trefoils. In particular, unlike most knot invariants,we show that the spherical stick index distinguishes between the granny and square knots, and that composing a nontrivial knot with a second nontrivial knot need not increase its spherical stick index. 1. Introduction The stick index s[K] of a knot type [K] is the smallest number of straight line segments required to create a polygonal conformation of [K] in space. The stick index is generally difficult to compute. However, stick indices of small crossing knots are known, and stick indices for certain infinite categories of knots have been determined: Theorem 1.1 ([Jin97]). If Tp;q is a (p; q)-torus knot with p < q < 2p, s[Tp;q] = 2q. Theorem 1.2 ([ABGW97]). If nT is a composition of n trefoils, s[nT ] = 2n + 4. Despite the interest in stick index, two-dimensional analogues have not been studied in depth.
  • Knots and Links in Lens Spaces

    Knots and Links in Lens Spaces

    Alma Mater Studiorum Università di Bologna Dottorato di Ricerca in MATEMATICA Ciclo XXVI Settore Concorsuale di afferenza: 01/A2 Settore Scientifico disciplinare: MAT/03 Knots and links in lens spaces Tesi di Dottorato presentata da: Enrico Manfredi Coordinatore Dottorato: Relatore: Prof.ssa Prof. Giovanna Citti Michele Mulazzani Esame Finale anno 2014 Contents Introduction 1 1 Representation of lens spaces 9 1.1 Basic definitions . 10 1.2 A lens model for lens spaces . 11 1.3 Quotient of S3 model . 12 1.4 Genus one Heegaard splitting model . 14 1.5 Dehn surgery model . 15 1.6 Results about lens spaces . 17 2 Links in lens spaces 19 2.1 General definitions . 19 2.2 Mixed link diagrams . 22 2.3 Band diagrams . 23 2.4 Grid diagrams . 25 3 Disk diagram and Reidemeister-type moves 29 3.1 Disk diagram . 30 3.2 Generalized Reidemeister moves . 32 3.3 Standard form of the disk diagram . 36 3.4 Connection with band diagram . 38 3.5 Connection with grid diagram . 42 4 Group of links in lens spaces via Wirtinger presentation 47 4.1 Group of the link . 48 i ii CONTENTS 4.2 First homology group . 52 4.3 Relevant examples . 54 5 Twisted Alexander polynomials for links in lens spaces 57 5.1 The computation of the twisted Alexander polynomials . 57 5.2 Properties of the twisted Alexander polynomials . 59 5.3 Connection with Reidemeister torsion . 61 6 Lifting links from lens spaces to the 3-sphere 65 6.1 Diagram for the lift via disk diagrams . 66 6.2 Diagram for the lift via band and grid diagrams .
  • Introduction to Vassiliev Knot Invariants First Draft. Comments

    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.
  • CALIFORNIA STATE UNIVERSITY, NORTHRIDGE P-Coloring Of

    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.