Oriented Pair (S 3,S1); Two Knots Are Regarded As
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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. -
MUTATIONS of LINKS in GENUS 2 HANDLEBODIES 1. Introduction
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 1, January 1999, Pages 309{314 S 0002-9939(99)04871-6 MUTATIONS OF LINKS IN GENUS 2 HANDLEBODIES D. COOPER AND W. B. R. LICKORISH (Communicated by Ronald A. Fintushel) Abstract. A short proof is given to show that a link in the 3-sphere and any link related to it by genus 2 mutation have the same Alexander polynomial. This verifies a deduction from the solution to the Melvin-Morton conjecture. The proof here extends to show that the link signatures are likewise the same and that these results extend to links in a homology 3-sphere. 1. Introduction Suppose L is an oriented link in a genus 2 handlebody H that is contained, in some arbitrary (complicated) way, in S3.Letρbe the involution of H depicted abstractly in Figure 1 as a π-rotation about the axis shown. The pair of links L and ρL is said to be related by a genus 2 mutation. The first purpose of this note is to prove, by means of long established techniques of classical knot theory, that L and ρL always have the same Alexander polynomial. As described briefly below, this actual result for knots can also be deduced from the recent solution to a conjecture, of P. M. Melvin and H. R. Morton, that posed a problem in the realm of Vassiliev invariants. It is impressive that this simple result, readily expressible in the language of the classical knot theory that predates the Jones polynomial, should have emerged from the technicalities of Vassiliev invariants. -
The Conway Knot Is Not Slice
THE CONWAY KNOT IS NOT SLICE LISA PICCIRILLO Abstract. A knot is said to be slice if it bounds a smooth properly embedded disk in B4. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This com- pletes the classification of slice knots under 13 crossings, and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot. 1. Introduction The classical study of knots in S3 is 3-dimensional; a knot is defined to be trivial if it bounds an embedded disk in S3. Concordance, first defined by Fox in [Fox62], is a 4-dimensional extension; a knot in S3 is trivial in concordance if it bounds an embedded disk in B4. In four dimensions one has to take care about what sort of disks are permitted. A knot is slice if it bounds a smoothly embedded disk in B4, and topologically slice if it bounds a locally flat disk in B4. There are many slice knots which are not the unknot, and many topologically slice knots which are not slice. It is natural to ask how characteristics of 3-dimensional knotting interact with concordance and questions of this sort are prevalent in the literature. Modifying a knot by positive mutation is particularly difficult to detect in concordance; we define positive mutation now. A Conway sphere for an oriented knot K is an embedded S2 in S3 that meets the knot 3 transversely in four points. The Conway sphere splits S into two 3-balls, B1 and B2, and ∗ K into two tangles KB1 and KB2 . -
KNOTS and SEIFERT SURFACES Your Task As a Group, Is to Research the Topics and Questions Below, Write up Clear Notes As a Group
KNOTS AND SEIFERT SURFACES MATH 180, SPRING 2020 Your task as a group, is to research the topics and questions below, write up clear notes as a group explaining these topics and the answers to the questions, and then make a video presenting your findings. Your video and notes will be presented to the class to teach them your findings. Make sure that in your notes and video you give examples and intuition, along with formal definitions, theorems, proofs, or calculations. Make sure that you point out what the is most important take away message, and what aspects may be tricky or confusing to understand at first. You will need to work together as a group. Every member of the group must understand Problems 1,2, and 3, and each group member should solve at least one example from problem (4). You may split up Problems 5-9. 1. Resources The primary resource for this project is The Knot Book by Colin Adams, Chapter 4.3 (page 95-106). An Introduction to Knot Theory by Raymond Lickorish Chapter 2, could also be helpful. You may also look at other resources online about knot theory and Seifert surfaces. Make sure to cite the sources you use. 2. Topics and Questions As you research, you may find more examples, definitions, and questions, which you defi- nitely should feel free to include in your notes and/or video, but make sure you at least go through the following discussion and questions. (1) What is a Seifert surface for a knot? Describe Seifert's algorithm for finding Seifert surfaces: how do you find Seifert circles and how do you use those Seifert -
CROSSCAP NUMBERS of PRETZEL KNOTS 1. Introduction It Is Well
CROSSCAP NUMBERS OF PRETZEL KNOTS 市原 一裕 (KAZUHIRO ICHIHARA) 大阪産業大学教養部 (OSAKA SANGYO UNIVERSITY) 水嶋滋氏 (東京工業大学大学院情報理工学研究科) との共同研究 (JOINT WORK WITH SHIGERU MIZUSHIMA (TOKYO INSTITUTE OF TECHNOLOGY)) Abstract. For a non-trivial knot in the 3-sphere, the crosscap number is de¯ned as the minimal ¯rst betti number of non-orientable spanning surfaces for it. In this article, we report a simple formula of the crosscap number for pretzel knots. 1. Introduction It is well-known that any knot in the 3-sphere S3 bounds an orientable subsurface in S3; called a Seifert surface. One of the most basic invariant in knot theory, the genus of a knot K, is de¯ned to be the minimal genus of a Seifert surface for K. On the other hand, any knot in S3 also bounds a non-orientable subsurface in S3: Consider checkerboard surfaces for a diagram of the knot, one of which is shown to be non-orientable. In view of this, similarly as the genus of a knot, B.E. Clark de¯ned the crosscap number of a knot as follows. De¯nition (Clark, [1]). The crosscap number γ(K) of a knot K is de¯ned to be the minimal ¯rst betti number of a non-orientable surface spanning K in S3. For completeness we de¯ne γ(K) = 0 if and only if K is the unknot. Example. The ¯gure-eight knot K in S3 bounds a once-punctured Klein bottle S, appearing as a checkerboard surface for the diagram of K with minimal crossings. Thus γ(K) · ¯1(S) = 2. -
Knots of Genus One Or on the Number of Alternating Knots of Given Genus
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 7, Pages 2141{2156 S 0002-9939(01)05823-3 Article electronically published on February 23, 2001 KNOTS OF GENUS ONE OR ON THE NUMBER OF ALTERNATING KNOTS OF GIVEN GENUS A. STOIMENOW (Communicated by Ronald A. Fintushel) Abstract. We prove that any non-hyperbolic genus one knot except the tre- foil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number. 1. Introduction The motivation for the present paper came out of considerations of Gauß di- agrams recently introduced by Polyak and Viro [23] and Fiedler [12] and their applications to positive knots [29]. For the definition of a positive crossing, positive knot, Gauß diagram, linked pair p; q of crossings (denoted by p \ q)see[29]. Among others, the Polyak-Viro-Fiedler formulas gave a new elegant proof that any positive diagram of the unknot has only reducible crossings. A \classical" argument rewritten in terms of Gauß diagrams is as follows: Let D be such a diagram. Then the Seifert algorithm must give a disc on D (see [9, 29]). Hence n(D)=c(D)+1, where c(D) is the number of crossings of D and n(D)thenumber of its Seifert circles. Therefore, smoothing out each crossing in D must augment the number of components. If there were a linked pair in D (that is, a pair of crossings, such that smoothing them both out according to the usual skein rule, we obtain again a knot rather than a three component link diagram) we could choose it to be smoothed out at the beginning (since the result of smoothing out all crossings in D obviously is independent of the order of smoothings) and smoothing out the second crossing in the linked pair would reduce the number of components. -
Knot Genus Recall That Oriented Surfaces Are Classified by the Either Euler Characteristic and the Number of Boundary Components (Assume the Surface Is Connected)
Knots and how to detect knottiness. Figure-8 knot Trefoil knot The “unknot” (a mathematician’s “joke”) There are lots and lots of knots … Peter Guthrie Tait Tait’s dates: 1831-1901 Lord Kelvin (William Thomson) ? Knots don’t explain the periodic table but…. They do appear to be important in nature. Here is some knotted DNA Science 229, 171 (1985); copyright AAAS A 16-crossing knot one of 1,388,705 The knot with archive number 16n-63441 Image generated at http: //knotilus.math.uwo.ca/ A 23-crossing knot one of more than 100 billion The knot with archive number 23x-1-25182457376 Image generated at http: //knotilus.math.uwo.ca/ Spot the knot Video: Robert Scharein knotplot.com Some other hard unknots Measuring Topological Complexity ● We need certificates of topological complexity. ● Things we can compute from a particular instance of (in this case) a knot, but that does change under deformations. ● You already should know one example of this. ○ The linking number from E&M. What kind of tools do we need? 1. Methods for encoding knots (and links) as well as rules for understanding when two different codings are give equivalent knots. 2. Methods for measuring topological complexity. Things we can compute from a particular encoding of the knot but that don’t agree for different encodings of the same knot. Knot Projections A typical way of encoding a knot is via a projection. We imagine the knot K sitting in 3-space with coordinates (x,y,z) and project to the xy-plane. We remember the image of the projection together with the over and under crossing information as in some of the pictures we just saw. -
Visualization of Seifert Surfaces Jarke J
IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 1, NO. X, AUGUST 2006 1 Visualization of Seifert Surfaces Jarke J. van Wijk Arjeh M. Cohen Technische Universiteit Eindhoven Abstract— The genus of a knot or link can be defined via Oriented surfaces whose boundaries are a knot K are called Seifert surfaces. A Seifert surface of a knot or link is an Seifert surfaces of K, after Herbert Seifert, who gave an algo- oriented surface whose boundary coincides with that knot or rithm to construct such a surface from a diagram describing the link. Schematic images of these surfaces are shown in every text book on knot theory, but from these it is hard to understand knot in 1934 [13]. His algorithm is easy to understand, but this their shape and structure. In this article the visualization of does not hold for the geometric shape of the resulting surfaces. such surfaces is discussed. A method is presented to produce Texts on knot theory only contain schematic drawings, from different styles of surface for knots and links, starting from the which it is hard to capture what is going on. In the cited paper, so-called braid representation. Application of Seifert's algorithm Seifert also introduced the notion of the genus of a knot as leads to depictions that show the structure of the knot and the surface, while successive relaxation via a physically based the minimal genus of a Seifert surface. The present article is model gives shapes that are natural and resemble the familiar dedicated to the visualization of Seifert surfaces, as well as representations of knots. -
Grid Homology for Knots and Links
Grid Homology for Knots and Links Peter S. Ozsvath, Andras I. Stipsicz, and Zoltan Szabo This is a preliminary version of the book Grid Homology for Knots and Links published by the American Mathematical Society (AMS). This preliminary version is made available with the permission of the AMS and may not be changed, edited, or reposted at any other website without explicit written permission from the author and the AMS. Contents Chapter 1. Introduction 1 1.1. Grid homology and the Alexander polynomial 1 1.2. Applications of grid homology 3 1.3. Knot Floer homology 5 1.4. Comparison with Khovanov homology 7 1.5. On notational conventions 7 1.6. Necessary background 9 1.7. The organization of this book 9 1.8. Acknowledgements 11 Chapter 2. Knots and links in S3 13 2.1. Knots and links 13 2.2. Seifert surfaces 20 2.3. Signature and the unknotting number 21 2.4. The Alexander polynomial 25 2.5. Further constructions of knots and links 30 2.6. The slice genus 32 2.7. The Goeritz matrix and the signature 37 Chapter 3. Grid diagrams 43 3.1. Planar grid diagrams 43 3.2. Toroidal grid diagrams 49 3.3. Grids and the Alexander polynomial 51 3.4. Grid diagrams and Seifert surfaces 56 3.5. Grid diagrams and the fundamental group 63 Chapter 4. Grid homology 65 4.1. Grid states 65 4.2. Rectangles connecting grid states 66 4.3. The bigrading on grid states 68 4.4. The simplest version of grid homology 72 4.5. -
The Ways I Know How to Define the Alexander Polynomial
ALL THE WAYS I KNOW HOW TO DEFINE THE ALEXANDER POLYNOMIAL KYLE A. MILLER Abstract. These began as notes for a talk given at the Student 3-manifold seminar, Spring 2019. There seems to be many ways to define the Alexander polynomial, all of which are somehow interrelated, but sometimes there is not an obvious path between any two given definitions. As the title suggests, this is an exploration of all the ways I know how to define this knot invariant. While we will touch on a number of facts and properties, these notes are not meant to be a complete survey of the Alexander polynomial. Contents 1. Introduction2 2. Alexander’s definition2 2.1. The Dehn presentation2 2.2. Abelianization4 2.3. The associated matrix4 3. The Alexander modules6 3.1. Orders7 3.2. Elementary ideals8 3.3. The Fox calculus 10 3.4. Torus knots 13 3.5. The Wirtinger presentation 13 3.6. Generalization to links 15 3.7. Fibered knots 15 3.8. Seifert presentation 16 3.9. Duality 18 4. The Conway potential 18 4.1. Alexander’s three-term relation 20 5. The HOMFLY-PT polynomial 24 6. Kauffman state sum 26 6.1. Another state sum 29 7. The Burau representation 29 8. Vassiliev invariants 29 9. The Alexander quandle 29 9.1. Projective Alexander quandles 33 10. Reidemeister torsion 33 11. Knot Floer homology 33 References 33 Date: May 3, 2019. 1 DRAFT 2020/11/14 18:57:28 2 KYLE A. MILLER 1. Introduction Recall that a link is an embedded closed 1-manifold in S3, and a knot is a 1-component link. -
Lectures Notes on Knot Theory
Lectures notes on knot theory Andrew Berger (instructor Chris Gerig) Spring 2016 1 Contents 1 Disclaimer 4 2 1/19/16: Introduction + Motivation 5 3 1/21/16: 2nd class 6 3.1 Logistical things . 6 3.2 Minimal introduction to point-set topology . 6 3.3 Equivalence of knots . 7 3.4 Reidemeister moves . 7 4 1/26/16: recap of the last lecture 10 4.1 Recap of last lecture . 10 4.2 Intro to knot complement . 10 4.3 Hard Unknots . 10 5 1/28/16 12 5.1 Logistical things . 12 5.2 Question from last time . 12 5.3 Connect sum operation, knot cancelling, prime knots . 12 6 2/2/16 14 6.1 Orientations . 14 6.2 Linking number . 14 7 2/4/16 15 7.1 Logistical things . 15 7.2 Seifert Surfaces . 15 7.3 Intro to research . 16 8 2/9/16 { The trefoil is knotted 17 8.1 The trefoil is not the unknot . 17 8.2 Braids . 17 8.2.1 The braid group . 17 9 2/11: Coloring 18 9.1 Logistical happenings . 18 9.2 (Tri)Colorings . 18 10 2/16: π1 19 10.1 Logistical things . 19 10.2 Crash course on the fundamental group . 19 11 2/18: Wirtinger presentation 21 2 12 2/23: POLYNOMIALS 22 12.1 Kauffman bracket polynomial . 22 12.2 Provoked questions . 23 13 2/25 24 13.1 Axioms of the Jones polynomial . 24 13.2 Uniqueness of Jones polynomial . 24 13.3 Just how sensitive is the Jones polynomial . -
Concordance and Mutation
ISSN 1364-0380 (on line) 1465-3060 (printed) 831 eometry & opology G T Volume 5 (2001) 831–883 Published: 21 November 2001 Concordance and Mutation P Kirk and C Livingston Department of Mathematics Indiana University, Bloomington, IN 47405, USA Email: [email protected], [email protected] Abstract We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two. The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n> 1 there is an isomorphism φ from the concordance group n 2n+1 C of knotted (2n 1)–spheres in S to an algebraically defined group ± ; further- − ∞ ∞ ∞ G more, ± is isomorphic to the infinite direct sum Z Z2 Z4 . It was a startling consequenceG of the work of Casson and Gordon that in⊕ the classica⊕ l case the kernel of φ on 1 is infinitely generated. Beyond this, little has been discovered about the pair C ( 1, φ). C In this paper we present a new approach to studying 1 by introducing a group, , defined as the quotient of the set of knots by the equivalenceC relation generatedM by concordance and positive mutation, with group operation induced by connected sum. φ1 φ2 We prove there is a factorization of φ, 1 − . Our main result is that both maps have infinitely generated kernels. C −→M−→G Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials.