Knots and Links Derived from Prismatic Graphs
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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 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Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 15 August 2018 doi:10.20944/preprints201808.0265.v1 Peer-reviewed version available at Symmetry 2018, 10, 604; doi:10.3390/sym10110604 Article A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. -
Absolutely Exotic Compact 4-Manifolds
ABSOLUTELY EXOTIC COMPACT 4-MANIFOLDS 1 2 SELMAN AKBULUT AND DANIEL RUBERMAN Abstract. We show how to construct absolutely exotic smooth struc- tures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent to W that are absolutely exotic copies of each other. In this context, absolute means that the exotic structure is not relative to a particular parame- terization of the boundary. Our examples are constructed by modifying a relatively exotic manifold by adding an invertible homology cobor- dism along its boundary. Applying this technique to corks (contractible manifolds with a diffeomorphism of the boundary that does not extend to a diffeomorphism of the interior) gives examples of absolutely exotic smooth structures on contractible 4-manifolds. 1. Introduction One goal of 4-dimensional topology is to find exotic smooth structures on the simplest of closed 4-manifolds, such as S4 and CP2. Amongst mani- folds with boundary, there are very simple exotic structures coming from the phenomenon of corks, which are relatively exotic contractible mani- arXiv:1410.1461v3 [math.GT] 11 Dec 2014 folds discovered by the first-named author [2]. More specifically, a cork is a compact smooth contractible manifold W together with a diffeomor- phism f : ∂W → ∂W which does not extend to a self-diffeomorphism of W , although it does extend to a self-homeomorphism F : W → W . This gives an exotic smooth structure on W relative to its boundary, namely the pullback smooth structure by F. -
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 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. -
Knot Cobordisms, Bridge Index, and Torsion in Floer Homology
KNOT COBORDISMS, BRIDGE INDEX, AND TORSION IN FLOER HOMOLOGY ANDRAS´ JUHASZ,´ MAGGIE MILLER AND IAN ZEMKE Abstract. Given a connected cobordism between two knots in the 3-sphere, our main result is an inequality involving torsion orders of the knot Floer homology of the knots, and the number of local maxima and the genus of the cobordism. This has several topological applications: The torsion order gives lower bounds on the bridge index and the band-unlinking number of a knot, the fusion number of a ribbon knot, and the number of minima appearing in a slice disk of a knot. It also gives a lower bound on the number of bands appearing in a ribbon concordance between two knots. Our bounds on the bridge index and fusion number are sharp for Tp;q and Tp;q#T p;q, respectively. We also show that the bridge index of Tp;q is minimal within its concordance class. The torsion order bounds a refinement of the cobordism distance on knots, which is a metric. As a special case, we can bound the number of band moves required to get from one knot to the other. We show knot Floer homology also gives a lower bound on Sarkar's ribbon distance, and exhibit examples of ribbon knots with arbitrarily large ribbon distance from the unknot. 1. Introduction The slice-ribbon conjecture is one of the key open problems in knot theory. It states that every slice knot is ribbon; i.e., admits a slice disk on which the radial function of the 4-ball induces no local maxima. -
Introduction to Vassiliev Knot Invariants S
Cambridge University Press 978-1-107-02083-2 — Introduction to Vassiliev Knot Invariants S. Chmutov , S. Duzhin , J. Mostovoy Excerpt More Information 1 Knots and their relatives This book is about knots. It is, however, hardly possible to speak about knots without mentioning other one-dimensional topological objects embedded into the three-dimensional space. Therefore, in this introductory chapter we give basic def- initions and constructions pertaining to knots and their relatives: links, braids and tangles. The table of knots provided in Table 1.1 will be used throughout the book as a source of examples and exercises. 1.1 Definitions and examples 1.1.1 Knots A knot is a closed non-self-intersecting curve in 3-space. In this book, we shall mainly study smooth oriented knots. A precise definition can be given as follows. Definition 1.1. A parametrized knot is an embedding of the circle S1 into the Euclidean space R3. Recall that an embedding is a smooth map which is injective and whose differ- ential is nowhere zero. In our case, the non-vanishing of the differential means that the tangent vector to the curve is non-zero. In the above definition and everywhere in the sequel, the word smooth means infinitely differentiable. A choice of an orientation for the parametrizing circle S1 ={(cos t, sin t) | t ∈ R}⊂R2 gives an orientation to all the knots simultaneously. We shall always assume that S1 is oriented counterclockwise. We shall also fix an orientation of the 3-space; each time we pick a basis for R3 we shall assume that it is consistent with the orientation. -
An Odd Analog of Plamenevskaya's Invariant of Transverse
AN ODD ANALOG OF PLAMENEVSKAYA'S INVARIANT OF TRANSVERSE KNOTS by GABRIEL MONTES DE OCA A DISSERTATION Presented to the Department of Mathematics and the Graduate School of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy September 2020 DISSERTATION APPROVAL PAGE Student: Gabriel Montes de Oca Title: An Odd Analog of Plamenevskaya's Invariant of Transverse Knots This dissertation has been accepted and approved in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Department of Mathematics by: Robert Lipshitz Chair Daniel Dugger Core Member Nicholas Proudfoot Core Member Micah Warren Core Member James Brau Institutional Representative and Kate Mondloch Interim Vice Provost and Dean of the Graduate School Original approval signatures are on file with the University of Oregon Graduate School. Degree awarded September 2020 ii c 2020 Gabriel Montes de Oca iii DISSERTATION ABSTRACT Gabriel Montes de Oca Doctor of Philosophy Department of Mathematics September 2020 Title: An Odd Analog of Plamenevskaya's Invariant of Transverse Knots Plamenevskaya defined an invariant of transverse links as a distinguished class in the even Khovanov homology of a link. We define an analog of Plamenevskaya's invariant in the odd Khovanov homology of Ozsv´ath,Rasmussen, and Szab´o. We show that the analog is also an invariant of transverse links and has similar properties to Plamenevskaya's invariant. We also show that the analog invariant can be identified with an equivalent invariant in the reduced odd Khovanov homology. We demonstrate computations of the invariant on various transverse knot pairs with the same topological knot type and self-linking number. -
Bracket Calculations -- Pdf Download
Knot Theory Notes - Detecting Links With The Jones Polynonmial by Louis H. Kau®man 1 Some Elementary Calculations The formula for the bracket model of the Jones polynomial can be indicated as follows: The letter chi, Â, denotes a crossing in a link diagram. The barred letter denotes the mirror image of this ¯rst crossing. A crossing in a diagram for the knot or link is expanded into two possible states by either smoothing (reconnecting) the crossing horizontally, , or 2 2 vertically ><. Any closed loop (without crossings) in the plane has value ± = A ³A¡ : ¡ ¡ 1  = A + A¡ >< ³ 1  = A¡ + A >< : ³ One useful consequence of these formulas is the following switching formula 1 2 2 A A¡  = (A A¡ ) : ¡ ¡ ³ Note that in these conventions the A-smoothing of  is ; while the A-smoothing of  is >< : Properly interpreted, the switching formula abo³ve says that you can switch a crossing and smooth it either way and obtain a three diagram relation. This is useful since some computations will simplify quite quickly with the proper choices of switching and smoothing. Remember that it is necessary to keep track of the diagrams up to regular isotopy (the equivalence relation generated by the second and third Reidemeister moves). Here is an example. View Figure 1. K U U' Figure 1 { Trefoil and Two Relatives You see in Figure 1, a trefoil diagram K, an unknot diagram U and another unknot diagram U 0: Applying the switching formula, we have 1 2 2 A¡ K AU = (A¡ A )U 0 ¡ ¡ 3 3 2 6 and U = A and U 0 = ( A¡ ) = A¡ : Thus ¡ ¡ 1 3 2 2 6 A¡ K A( A ) = (A¡ A )A¡ : ¡ ¡ ¡ Hence 1 4 8 4 A¡ K = A + A¡ A¡ : ¡ ¡ Thus 5 3 7 K = A A¡ + A¡ : ¡ ¡ This is the bracket polynomial of the trefoil diagram K: We have used to same symbol for the diagram and for its polynomial. -
On Cosmetic Surgery on Knots K
Cosmetic surgery on knots On cosmetic surgery on knots K. Ichihara Introduction 3-manifold Dehn surgery Cosmetic Surgery Kazuhiro Ichihara Conjecture Known Facts Examples Nihon University Criteria College of Humanities and Sciences Results (1) Jones polynomial 2-bridge knot Hanselman's result Finite type invariants Based on Joint work with Recent progress Tetsuya Ito (Kyoto Univ.), In Dae Jong (Kindai Univ.), Results (2) Genus one alternating knots Thomas Mattman (CSU, Chico), Toshio Saito (Joetsu Univ. of Edu.), Cosmetic Banding and Zhongtao Wu (CUHK) How To Check? The 67th Topology Symposium - Online, MSJ. Oct 13, 2020. 1 / 36 Cosmetic surgery Papers on knots I (with Toshio Saito) K. Ichihara Cosmetic surgery and the SL(2; C) Casson invariant for 2-bridge knots. Introduction 3-manifold Hiroshima Math. J. 48 (2018), no. 1, 21-37. Dehn surgery Cosmetic Surgery I (with In Dae Jong (Appendix by Hidetoshi Masai)) Conjecture Known Facts Cosmetic banding on knots and links. Examples Osaka J. Math. 55 (2018), no. 4, 731-745. Criteria Results (1) I (with Zhongtao Wu) Jones polynomial 2-bridge knot A note on Jones polynomial and cosmetic surgery. Hanselman's result Finite type invariants Comm. Anal. Geom. 27 (2019), no.5, 1087{1104. Recent progress I (with Toshio Saito and Tetsuya Ito) Results (2) Genus one alternating Chirally cosmetic surgeries and Casson invariants. knots Cosmetic Banding To appear in Tokyo J. Math. How To Check? I (with In Dae Jong, Thomas W. Mattman, Toshio Saito) Two-bridge knots admit no purely cosmetic surgeries. To appear in Algebr. Geom. Topol. 2 / 36 Table of contents Introduction Jones polynomial 3-manifold 2-bridge knot Dehn surgery Hanselman's result Finite type invariants Cosmetic Surgery Conjecture Recent progress Known Facts Results (2) [Chirally cosmetic surgery] Examples Alternating knot of genus one Criteria Cosmetic Banding Results (1) [Purely cosmetic surgery] How To Check? Cosmetic surgery Classification of 3-manifolds on knots K. -
Knots, Molecules, and the Universe: an Introduction to Topology
KNOTS, MOLECULES, AND THE UNIVERSE: AN INTRODUCTION TO TOPOLOGY AMERICAN MATHEMATICAL SOCIETY https://doi.org/10.1090//mbk/096 KNOTS, MOLECULES, AND THE UNIVERSE: AN INTRODUCTION TO TOPOLOGY ERICA FLAPAN with Maia Averett David Clark Lew Ludwig Lance Bryant Vesta Coufal Cornelia Van Cott Shea Burns Elizabeth Denne Leonard Van Wyk Jason Callahan Berit Givens Robin Wilson Jorge Calvo McKenzie Lamb Helen Wong Marion Moore Campisi Emille Davie Lawrence Andrea Young AMERICAN MATHEMATICAL SOCIETY 2010 Mathematics Subject Classification. Primary 57M25, 57M15, 92C40, 92E10, 92D20, 94C15. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-96 Library of Congress Cataloging-in-Publication Data Flapan, Erica, 1956– Knots, molecules, and the universe : an introduction to topology / Erica Flapan ; with Maia Averett [and seventeen others]. pages cm Includes index. ISBN 978-1-4704-2535-7 (alk. paper) 1. Topology—Textbooks. 2. Algebraic topology—Textbooks. 3. Knot theory—Textbooks. 4. Geometry—Textbooks. 5. Molecular biology—Textbooks. I. Averett, Maia. II. Title. QA611.F45 2015 514—dc23 2015031576 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. -
On the Links–Gould Invariant of Links
On the Links–Gould Invariant of Links David De Wit,∗ Louis H Kauffman† and Jon R Links∗ Abstract We introduce and study in detail an invariant of (1, 1) tangles. This invariant, de- rived from a family of four dimensional representations of the quantum superalgebra Uq[gl(2|1)], will be referred to as the Links–Gould invariant. We find that our in- variant is distinct from the Jones, HOMFLY and Kauffman polynomials (detecting chirality of some links where these invariants fail), and that it does not distinguish mutants or inverses. The method of evaluation is based on an abstract tensor state model for the invariant that is quite useful for computation as well as theoretical exploration. arXiv:math/9811128v2 [math.GT] 24 Nov 1998 1 Introduction Since the discovery of the Jones polynomial [14], several new invariants of knots, links and tangles have become available due to the development of sophisticated mathematical techniques. Among these, the quantum algebras as defined by Drinfeld [9] and Jimbo [13], being examples of quasi-triangular Hopf algebras, provide a systematic means of solving the Yang–Baxter equation and in turn may be employed to construct representations of the braid group. From each of these representations, a prescription exists to compute invariants of oriented knots and links [34, 39, 41], from which the Jones polynomial is recoverable using the simplest quantum algebra Uq [sl(2)] in its minimal (2-dimensional) representation. ∗David De Wit and Jon R Links: Department of Mathematics, The University of Queensland. Q, 4072, Australia. email: [email protected], [email protected].