DOCSLIB.ORG

Knot Invariants—Coloring We'll Develop a Few Tools For

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Knot Invariants—Coloring We'll Develop a Few Tools For Knot Invariants|Coloring We'll develop a few tools for distinguishing knots, including the simple, yet useful, coloring invariants. It's generally too hard to tell when two knots are the same or not by looking at their diagrams. In fact, if you're lucky you can prove they are the same, but you can never prove they are different because there are infinitely many ways you can alter the diagram through Reidemeister moves, and you can never know if you just missed the route to get from one to another. For example, to start to study knots, people started making table of \prime" knots by drawing all possible diagrams with a given number of crossings and using Reidemeister to try to figure out which were different. One of the main such projects was done by Tait and Little in the 1890's. They included these two knots: Unfortunately, these two are the same, though many people missed this fact|until 1973 when it was discovered by a lawyer named Kenneth Perko who did math for fun in his spare time. These two are now called the \Perko pair" even though they are really the same. So what we need to do is to find something that is easy to calculate from a diagram, but that does not change when using Reidemeister moves so that different versions of the same know possess the same property. Such a thing is called a knot invariant. Ideally we could calculate something simple and it could tell any two different knots apart. We haven't found anything yet, but who knows? Here is an example of a property that is easy to tell from a knot diagram, and doesn't change when we do Reidemeister moves. It's called 3-colorability and all it really asks is if we can paint the strands of a knot diagram according to certain rules. Definition: A knot is 3 colorable if we can paint the strands of its diagram using three different colors, with the stipulation that at any crossing, either all three strands must be the same color, or else all three must be different. We also require the use of more tha one color, because painting all the strands the same color is a trivial way to solve the problem! It is actually pretty obvious that the trefoil is 3-colorable, by painting each strand a different color. Note that at each of the three crossings, all three colors are used. What 1 happens if we were to try to paint the figure-of-eight with only three colors? Try! There's no way to make it work! This means that the trefoil and the figure-of-eight must be different knots. Before we say this for sure, let's verify that if one diagram for a knot is 3-colorable, then all of them are. This is easy for twisting. For poking, everything is OK if both strands are the same color. If they aren't, then the part that gets \poked under" the other part will acquire the third color, so again the result is 3-colorable. The slide move is a little trickier and breaks down into cases depending on how many different colors there are in the part of the diagram being examined. But going through the cases shows that, indeed, 3-colorability in invariant under Reidemeister moves. So it is a knot invariant. We could extend the definition to other numbers: Definition: A knot or link is n-colorable if the strands of a diagram can be painted with no more than n different colors (numbered 1{n, or more usually 0{n − 1), using at least two to avoid trivial monochromatic diagrams, and so that at each crossing the colors of the NW and SE strands add to the same amount, modulo n, as the colors of the SW and NE strands. The figure-of-eight knot happens to be 5-colorable, as is easily seen just by trying. We only need to check colorability for n a prime. Showing this will be a homework problem. Also, no knot is 2-colorable (although every link of two or more pieces is|just color one piece with one color, and the rest with a different color). Try the problems on the handout! 2.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Textile Research Journal
    Textile Research Journal http://trj.sagepub.com A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures S. Grishanov, V. Meshkov and A. Omelchenko Textile Research Journal 2009; 79; 822 DOI: 10.1177/0040517508096221 The online version of this article can be found at: http://trj.sagepub.com/cgi/content/abstract/79/9/822 Published by: http://www.sagepublications.com Additional services and information for Textile Research Journal can be found at: Email Alerts: http://trj.sagepub.com/cgi/alerts Subscriptions: http://trj.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.co.uk/journalsPermissions.nav Citations http://trj.sagepub.com/cgi/content/refs/79/9/822 Downloaded from http://trj.sagepub.com by Andrew Ranicki on October 11, 2009 Textile Research Journal Article A Topological Study of Textile Structures. Part II: Topological Invariants in Application to Textile Structures S. Grishanov1 Abstract This paper is the second in the series TEAM Research Group, De Montfort University, Leicester, on topological classification of textile structures. UK The classification problem can be resolved with the aid of invariants used in knot theory for classi- V. Meshkov and A. Omelchenko fication of knots and links. Various numerical and St Petersburg State Polytechnical University, St polynomial invariants are considered in application Petersburg, Russia to textile structures. A new Kauffman-type polyno- mial invariant is constructed for doubly-periodic textile structures. The values of the numerical and polynomial invariants are calculated for some sim- plest doubly-periodic interlaced structures and for some woven and knitted textiles.
    [Show full text]
  • An Exploration Into Knot Theory Application Regarding Enzymatic
    AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfllment Of the Requirements for the Degree Master of Science In Mathematics By Albert Joseph Jefferson IV 2019 SIGNATURE PAGE THESIS: AN EXPLORATION INTO KNOT THEORY APPLICATIONS REGARDING ENZYMATIC ACTION ON DNA MOLECULES AUTHOR: Albert Joseph Jefferson IV DATE SUBMITTED: Fall 2019 Department of Mathematics and Statistics Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. John Rock Mathematics & Statistics Dr. Amber Rosin Mathematics & Statistics ii ACKNOWLEDGMENTS I would frst like to thank my mother, who tirelessly continues to support every goal of mine, regardless of the feld, and whose guidance has been invaluable in my life. I also would like to thank my 3 sisters, each of whom are excellent role models and have given me so much inspiration. A giant thank you to my advisor, Robin Wilson, who has dealt with my procrastination gracefully, as well to my thesis committee for being so fexible with the defense (each of you are forever my mentors and I owe so much to you all). Finally, I’d like to thank my boyfriend who has been my rock through this entire process, I love you so so much. iii ABSTRACT This thesis will explore the applications of knot theory in biology at the graduate level, specifcally knot theory’s application on DNA super coiling, site specifc recombination, and the overall topology of DNA. We will start with a short introduction to knot theory, introduce rational tangles (the foundation for our tangle model for enzymatic reactions on DNA), review a very specifc overview of DNA and site specifc recombination as it pertains to our motivations, and fnally introduce the tangle model for a Tyrosine Recom- binase and its infuence on the topology of DNA following its action on the molecule.
    [Show full text]
  • On the Additivity of Crossing Numbers
    ON THE ADDITIVITY OF CROSSING NUMBERS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Alicia Arrua 2015 SIGNATURE PAGE THESIS: ON THE ADDITIVITY OF CROSSING NUMBERS AUTHOR: Alicia Arrua DATE SUBMITTED: Spring 2015 Mathematics and Statistics Department Dr. Robin Wilson Thesis Committee Chair Mathematics & Statistics Dr. Greisy Winicki-Landman Mathematics & Statistics Dr. Berit Givens Mathematics & Statistics ii ACKNOWLEDGMENTS This thesis would not have been possible without the invaluable knowledge and guidance from Dr. Robin Wilson. His support throughout this entire experience has been amazing and incredibly helpful. I’d also like to thank Dr. Greisy Winicki- Landman and Dr. Berit Givens for being a part of my thesis committee and offering their support. I’d also like to thank my family for putting up with my late nights of work and motivating me when I needed it. Lastly, thank you to the wonderful friends I’ve made during my time at Cal Poly Pomona, their humor and encouragement aided me more than they know. iii ABSTRACT The additivity of crossing numbers over a composition of links has been an open problem for over one hundred years. It has been proved that the crossing number over alternating links is additive independently in 1987 by Louis Kauffman, Kunio Murasugi, and Morwen Thistlethwaite. Further, Yuanan Diao and Hermann Gru­ ber independently proved that the crossing number is additive over a composition of torus links. In order to investigate the additivity of crossing numbers over a composition of a different class of links, we introduce a tool called the deficiency of a link.
    [Show full text]
  • Linking Numbers in Three-Manifolds
    LINKING NUMBERS IN THREE-MANIFOLDS PATRICIA CAHN AND ALEXANDRA KJUCHUKOVA Abstract. Let M be a connected, closed, oriented three-manifold and K, L two rationally null- homologous oriented simple closed curves in M. We give an explicit algorithm for computing the linking number between K and L in terms of a presentation of M as an irregular dihedral 3-fold cover of S3 branched along a knot α ⊂ S3. Since every closed, oriented three-manifold admits such a presentation, our results apply to all (well-defined) linking numbers in all three-manifolds. Furthermore, ribbon obstructions for a knot α can be derived from dihedral covers of α. The linking numbers we compute are necessary for evaluating one such obstruction. This work is a step toward testing potential counter-examples to the Slice-Ribbon Conjecture, among other applications. 1. Introduction The notion of linking number between knots in S3 dates back at least as far as Gauss [13]. More generally, given a closed, oriented three-manifold M and two rationally null-homologous, oriented, simple closed curves K; L ⊂ M, the linking number lk(K; L) is defined as well. It is given by 1 (K · C ), where C is a 2-chain in M with boundary nL, n 2 , and · denotes the signed n L L N intersection number. This linking number is well-defined and symmetric [3]. Let the three-manifold M be presented as a 3-fold irregular dihedral branched cover of S3, branched along a knot. Every closed oriented three-manifold admits such a presentation [16, 17, 21].
    [Show full text]
  • Cameron Mca. Gordon UT Austin, CNS September 19, 2018
    1 KNOTS Cameron McA. Gordon UT Austin, CNS September 19, 2018 2 A knot is a closed loop in space. 3 A knot is a closed loop in space. unknot 4 c(K)= crossing number of K = minimum number of crossings in any diagram of K. 5 c(K)= crossing number of K = minimum number of crossings in any diagram of K. 6 Vortex Atoms (Lord Kelvin, 1867) c(K) = order of knottiness of K Peter Guthrie Tait (1831-1901) 7 8 9 10 c(K) # of knots c(K) # of knots 3 1 11 552 4 1 12 2, 176 5 2 13 9, 988 6 3 14 46, 972 7 7 15 253, 293 8 21 16 1, 388, 705 9 49 17 8, 053, 378 10 165 11 c(K) # of knots c(K) # of knots 3 1 11 552 4 1 12 2, 176 5 2 13 9, 988 6 3 14 46, 972 7 7 15 253, 293 8 21 16 1, 388, 705 9 49 17 8, 053, 378 10 165 9, 755, 313 prime knots with c(K) ≤ 17. 12 How can you prove that two knots are different? 13 How can you prove that two knots are different? The Perko Pair 14 How can you prove that two knots are different? The Perko Pair How can you determine whether a given knot is the unknot or not? 15 There are many knot invariants that help with these questions. 16 There are many knot invariants that help with these questions. Example. Alexander polynomial (1928) 17 There are many knot invariants that help with these questions.
    [Show full text]
  • An Introduction to Knot Theory
    Compiled from presentations and work by: David Freund, Sarah Smith-Polderman, Danielle Shepherd, Joseph Smith, Michael Bush, and Katelyn French Advised by: Dr. John Ramsay and Dr. Jennifer Bowen Take a piece of string Loop it around and through itself Connect the ends A mathematical knot [I]. Links are composed of interwoven knots (components). A link with one component is a knot. Links are commonly represented as two- dimensional images called projections. Knot projections are different representations of a given knot Three projections of the trefoil knot [II]. The three Reidemeister moves [III]. In the mirror image of a link, every overcrossing is switched an undercrossing and vice versa Most knots, like the trefoil knot, are not amphichiral (equivalent to their mirror image) The left handed trefoil (left) and its mirror image, the right handed trefoil (right). There is no combination of Reidemeister moves that can transform one into the other [IV]. Knots and links can be classified through their invariants An invariant is an inherent characteristic of a link that is the same for any projection The Perko Pair, they were thought to be different knots for 75 years [V]! Crossing number P-colorability Linking number Knot groups Genus Bridge number Polynomials Braid Index The crossing number of a knot or link A, denoted c(A), is the fewest number of crossings that occurs in any projection of the knot or link. The crossing number of the trefoil is 3. The two projections on the left are at the trefoil’s crossing number, while the one on the right has 4 crossings [2].
    [Show full text]
  • All Tangled up in Knots
    All tangled up in knots Ryan Hansen Motivation for this talk • Understanding some foundational concepts • Differentiate between some knots • Present a few interesting (and sometimes a little surprising) results • State a few open problems • Mention applications What is a knot or link? • Knot - simple closed curve in space • Link - disjoint union of knots What is a projection? Right-handed trefoil Unknot Left-handed trefoil Figure-8 knot Figure-8 knot Figure-8 knot Reidemeister moves and ambient isotopy Unknot vs Trefoil • Wolfgang Haken, 1961 Open Problem: Write a computer program impletenting Haken’s algorithm • Hass and Lagrias - 2^(1,000,000,000n) In case you needed more convincing... • Perko Pair Some invariants • Crossing number, c(K) • Unknotting number, u(K) • Tricolorability Generalization of tricolorability • p-colorability for a prime p 2x y + z mod p ≡ Another example: Open problems for p-colorability • Is there a relationship between c(K) and the largest prime that admits a p-coloring? • If K is p-colorable for what q is K q-colorable (q=kp, but what others)? Algebraic Knots • Formed by closure of a tangle A simple tangle: More on tangles... • Addition L+M • Multiplication R Q Example: Surprising tangle test Continued fractions ? 232= 3 23 − ∼ − 1 2 12 12 3 1 2+ 1 =2+ = = =3 =3+ 1 3+ 2 5 5 5 − 5 2+ 3 − − Nakanishi’s Conjecture + Reidemeister Type I, II and III ? On a simple tangle 2= 5k 2 k Z − { − | ∈ } 1= 5k 1 k Z − { − | ∈ } 0= 5k k Z { | ∈ } 1= 5k +1 k Z { | ∈ } 2= 5k +2 k Z { | ∈ } ∞ Tangle algebra • Addition • Multiplication This shows: • Any algebraic link is (2,2)-equivalent to a link of two or fewer crossings Interesting • (2,2)-moves preserve 5-colorability • (p,q)-moves preserve certain colorabilities too Knot composition • Denoted K1#K2 • Many ways • Composite vs.
    [Show full text]
  • An Invariant of Regular Isotopy
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 318, Number 2, April 1990 AN INVARIANT OF REGULAR ISOTOPY LOUIS H. KAUFFMAN Abstract. This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted LK for a link K , and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. L0= 1. 3. ¿"y~ =aL' L—(5"~ = a~lL- 4- ¿a< +¿X. =2(¿s^ +L5C )• Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from L by writhe-normalization. I. Introduction In this paper I introduce a two variable Laurent polynomial invariant of knots and links in three-dimensional space. The primary version of this invariant, de- noted LK , is a regular isotopy invariant of unoriented knots and links. Regular isotopy will be explained below. Associated to LK are two normalized polynomials FK, for K oriented, and UK for K unoriented. These are each ambient isotopy invariants of K. Each is obtained from LK by multiplying it by a normalizing factor. Since FK is an invariant for oriented knots and links, it may be compared with the original Jones VK-polynomial [13] and with the homily polynomial PK [10]. In fact, FK has the Jones polynomial VK as a special case (see §3). FK is distinct from both the Jones and the homily polynomials. The jF-polynomial is good at distinguishing knots and links from their mirror images.
    [Show full text]
  • ADEQUACY of LINK FAMILIES Slavik Jablan, Ljiljana Radović , and Radmila Sazdanović
    PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 88(102) (2010), 21–52 DOI: 10.2298/PIM1002021J ADEQUACY OF LINK FAMILIES Slavik Jablan, Ljiljana Radović†, and Radmila Sazdanovi憆 Communicated by Rade Živaljević Abstract. We analyze adequacy of knots and links, utilizing Conway no- tation, Montesinos tangles and Linknot and KhoHo computer calculations. We introduce a numerical invariant called adequacy number, and compute adequacy polynomial which is the invariant of alternating link families. Ac- cording to computational results, adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links generated by links with at most 12 crossings. 1. Introduction We consider adequacy of nonalternating knots and links (shortly KLs) and their families (classes) given in the Conway notation. First, we explain the Conway no- tation, introduced in Conway’s seminal paper [1] published in 1967, and effectively used since (e.g., [2]). Conway symbols of knots with up to 10 crossings and links with at most 9 crossings are given in the Appendix of the book [3]. Figure 1. The elementary tangles. The main building blocks in the Conway notation are elementary tangles. We distinguish three elementary tangles, shown in Fig. 1 and denoted by 0, 1 and −1. All other tangles can be obtained by combining elementary tangles, while 0 and 1 are sufficient for generating alternating knots and links (abbr. KLs). Elementary 2000 Mathematics Subject Classification: 57M25, 57M27. Supported by the Serbian Ministry of Science and Technological Development, grant 144032D. 21 22 JABLAN, RADOVIĆ, AND SAZDANOVIĆ tangles can be combined by the following operations: sum, product,andramification (Figs.
    [Show full text]
  • An Introduction to Knot Theory Ortho Flint Orthofl[email protected]
    An Introduction to Knot Theory Ortho Flint orthofl[email protected] Go to: http://www.math.uwo.ca/∼ortho/website.html 1 Preliminaries A link L of m components is a subset of S3 that consists of m disjoint, piecewise linear simple closed curves, each of which is said to be a component of the link. A knot has one component. An 8 crossing knot 2 A diagram D of a link L is the projection onto a plane (or 2-sphere) of L such that every point of D is at most two points from L. If a point of D is two points from L then it is said to be a crossing of D, called a component crossing if both points belong to the same component of L, otherwise it is called a link crossing. Links L1 and L2 are said to be topologically equivalent if one can be transformed into the other using a smooth collision free motion. We say the diagrams for L1 and L2 are equivalent if the links are equivalent. It is conventional to indicate on a link diagram the additional information required to construct a link equivalent to the one from which the diagram was made. This information takes the form of over/under passes, indicated by a break in the image of the underpassing arc in a neighbourhood of the crossing. 3 We shall treat a link diagram as a 4-regular graph on S2 by considering each crossing as a vertex of the graph and the portion of the curve between two consecutive crossings as an edge between the two vertices.
    [Show full text]
  • KNOTS KNOTES Contents 1. Motivation, Basic Definitions And
    KNOTS KNOTES JUSTIN ROBERTS Contents 1. Motivation, basic definitions and questions 3 1.1. Basic definitions 3 1.2. Basic questions 4 1.3. Operations on knots 6 1.4. Alternating knots 7 1.5. Unknotting number 8 1.6. Further examples of knots and links 9 1.7. Methods 11 1.8. A table of the simplest knots and links 12 2. Formal definitions and Reidemeister moves 14 2.1. Knots and equivalence 14 2.2. Projections and diagrams 17 2.3. Reidemeister moves 18 2.4. Is there an algorithm for classifying and tabulating knots? 22 3. Simple invariants 25 3.1. Linking number 25 3.2. Invariants 27 3.3. 3-colourings 29 3.4. p-colourings 36 3.5. Optional: colourings and the Alexander polynomial. (Unfinished section!) 39 3.6. Some additional problems 43 4. The Jones polynomial 46 4.1. The Kauffman bracket 47 4.2. Correcting via the writhe 52 4.3. A state-sum model for the Kauffman bracket 53 4.4. The Jones polynomial and its skein relation 55 4.5. Some properties of the Jones polynomial 60 4.6. A characterisation of the Jones polynomial 61 4.7. How powerful is the Jones polynomial? 64 4.8. Alternating knots and the Jones polynomial 66 4.9. Other knot polynomials 71 5. A glimpse of manifolds 74 5.1. Metric spaces 74 5.2. Topological spaces 76 5.3. Hausdorff and second countable 78 5.4. Reconsidering the definition of n-manifold 79 6. Classification of surfaces 81 6.1. Combinatorial models for surfaces 81 Date: January 27, 2015.
    [Show full text]