DOCSLIB.ORG

Is the Jones Polynomial of a Knot Really a Polynomial?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Is the Jones Polynomial of a Knot Really a Polynomial? November 8, 2006 16:32 WSPC/134-JKTR 00491 Journal of Knot Theory and Its Ramifications Vol. 15, No. 8 (2006) 983–1000 c World Scientific Publishing Company IS THE JONES POLYNOMIAL OF A KNOT REALLY A POLYNOMIAL? STAVROS GAROUFALIDIS∗ and THANG TQ LEˆ † School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA ∗[email protected] †[email protected] Accepted 19 June 2006 Dedicated to Louis Kauffman on the occasion of his 60th birthday ABSTRACT The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro’s work on the colored Jones function. Keywords: Knots; Jones polynomial; Kauffman bracket; Habiro ring; homology spheres; TQFT; holonomic functions; A-polynomial; AJ Conjecture. Mathematics Subject Classification 2000: Primary 57N10; Secondary 57M25 J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com 1. Introduction 1.1. The Jones polynomial of a knot In 1985 Jones discovered the celebrated Jones polynomial of a knot/link in 3-space, see [14]. The framed version of the Jones polynomial of a framed oriented knot/link may be uniquely defined by the following skein theory: by MAX-PLANCK-INSTITUTE FOR MATHEMATICS LIBRARY on 03/17/20. Re-use and distribution is strictly not permitted, except for Open Access articles. From the above definition, the Jones polynomial JL of an oriented link L lies in Z[q±1/4]. 983 November 8, 2006 16:32 WSPC/134-JKTR 00491 984 S. Garoufalidis & T. T. Lˆe An important and trivial observation is that knots may be considered to lie in closed 3-manifolds other than S3. The paper is centered around the following question: Question 1.1. What is the Jones polynomial of a knot K in a closed 3-manifold N? In particular, is the Jones polynomial of a knot in a closed 3-manifold really a polynomial? Before we reveal the answer, let us review some alternative definitions of the Jones polynomial, namely the Kauffman bracket approach, the quantum group approach, the TQFT (Witten–Reshetikhin–Turaev) approach, and the perturbative TQFT approach. Note that being polynomial is closely related to the integrality discussed in [21]. 1.2. The Kauffman bracket of a knot Soon after Jones’s discovery, Kauffman gave a reformulation of the Jones polynomial in terms of the Kauffman bracket · : Framed unoriented links in S3 → Z[A±1]. The Kauffman bracket is defined by the following skein theory (see [15]): (1.1) and the relation between the Kauffman bracket and the Jones polynomial is the following: if L is an oriented m-component link projection with writhe w(L), then m 3 −w(L) JL|q1/4=A =(−1) (−A ) L. Since the Kauffman bracket takes values in Z[A±1], it implies once again that the Jones polynomial of an oriented knot/link in 3-space takes values in Z[q±1/4]. The Kauffman bracket can be generalized to framed unoriented links in a closed 3-manifold N and the resulting Kauffman bracket skein module S(N) can be defined to be the quotient of the free Z[A±1]-module on the set of framed unoriented links J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com in N, modulo the relations (1.1): S(N)=Z[A±1]-linear combinations of framed unoriented links in N/(relations (1.1)). From this point of view, the Kauffman bracket skein module of S(S3)isafree Z[A±1] with basis the empty link. This explains why the Jones polynomial of a link in S3 is an element of Z[A±1]. A knot in a general 3-manifold N gives, by definition, an element of the Kauffman bracket skein module S(N). However, little is known about the skein by MAX-PLANCK-INSTITUTE FOR MATHEMATICS LIBRARY on 03/17/20. Re-use and distribution is strictly not permitted, except for Open Access articles. module S(N). In particular, it is not known when it is free, nor is known a basis of it. In addition, S(N) is not a ring, but rather a module over the ring Z[A±1]. Thus, the skein module S(N) does not answer our Question 1.1. November 8, 2006 16:32 WSPC/134-JKTR 00491 Is the Jones Polynomial of a Knot Really a Polynomial? 985 1.3. The quantum group approach Using quantum group technology (see [25, 27]), one may define the colored Jones polynomial of a framed oriented knot in 3-space. Let us postpone the technical details to a later Sec. 2.1 and concentrate on a main idea: the notion of color. Suppose L is a framed r-component link in S3 with ordered components. The Jones polynomial of L is a powerful invariant that takes values in the Laurent polynomial ring R := Z[q±1/4]. (1.2) An even more powerful invariant is the colored Jones function r JL : N →R, which encodes the Jones polynomial of L together with its parallels. Here N is the set of positive integers, and JL(n1,...,nr)isthequantumsl2 invariant of the link L with colors the irreducible sl2-modules of dimensions n1,...,nr.When n1 = ···= nr = 2, the polynomial JL(n1,...,nr) is the Jones polynomial. Here we use the normalization such that when U is the unknot of 0 framing, qn/2 − q−n/2 JU (n)=[n]:= . q1/2 − q−1/2 1.4. The TQFT approach Let us fix a framed oriented knot K in a 3-manifold N and a nonnegative integer ∈ N ∈ Q n . The Witten–Reshetikhin–Turaev (WRT) invariant Z(N,K),ξ,Vn (ξ)is a generalization of the colored Jones polynomial and can be defined under the framework of topological quantum field theory (TQFT in short); see [27]. Here Vn is the n-dimensional irreducible representation of sl2 and ξ is a complex root of unity of order divisible by 4. When N = S3, and with the proper normalization, we may identify the WRT invariant with the colored Jones polynomial as follows: 3 Z(S ,K),ξ,Vn =evξJK (n), J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com where ±1/4 evξ : Z[q ] → Z[ξ] is the evaluation q1/4 = ξ. From the TQFT approach, it is not clear why the Jones polynomial is even a polynomial. 1.5. The perturbative approach pert by MAX-PLANCK-INSTITUTE FOR MATHEMATICS LIBRARY on 03/17/20. Re-use and distribution is strictly not permitted, except for Open Access articles. Perturbative quantum field theory also constructs an invariant Z(N,K),Vn which takes 3 pert Q 3 values in the power series ring [[h]]. When N = S , Z(S ,K),Vn is the composition of the Kontsevich integral of a knot with the sl2 weight system, and when N is November 8, 2006 16:32 WSPC/134-JKTR 00491 986 S. Garoufalidis & T. T. Lˆe pert sl2 arbitrary, Z(N,K),Vn is the composition of the LMO invariant with the weight system, see [1, 18]. When N = S3, the main identity is: pert h K ∈ Q ev J (n)=Z(N,K),Vn [[h]], where ±1/4 evh : Z[q ] → Q[[h]] is the evaluation q = eh; see, for example [22]. Thus, from the perturbative point of view, the Jones polynomial of a knot in a homology sphere exists as a formal power series in h. 1.6. The Habiro ring Despite the apparent failure of TQFT to explain the polynomial aspect of the Jones polynomial, there is one gain. Namely let us fix (N,K) a natural number n and let Ω4 denote the set of complex roots of unity whose order is divisible by 4. Then, the WRT invariant gives a function: → C Z(N,K),Vn :Ω4 . (1.3) This function is not continuous, and does not have nice analytic properties. Habiro introduced an alternative notion of analytic functions. The latter are by definition elements of the Habiro ring, defined by: 2 n Z[q] := lim Z[q]/((1 − q)(1 − q ) ···(1 − q )). (1.4) ←n The ring Z[q] can be considered as the set of all series of the form ∞ 2 n f(q)= fn(q)(1− q)(1 − q ) ···(1 − q ), wherefn(q) ∈ Z[q], (1.5) n=0 with the warning that f(q)doesnot uniquely determine (fn(q)). It turns out that the Habiro ring Z[q] shares many properties with the ring of germs of complex analytic functions, (see [9]) and plays an important role in J. Knot Theory Ramifications 2006.15:983-1000. Downloaded from www.worldscientific.com Quantum Topology. In particular, elements of the Habiro ring (a) can be differentiated with respect to q, (b) can be evaluated at the set Ω of complex roots of unity, (c) have Taylor series expansions that uniquely determine them, (d) form an integral domain.
Load more

Recommended publications
  • Topological Aspects of Biosemiotics
    tripleC 5(2): 49-63, 2007 ISSN 1726-670X http://tripleC.uti.at Topological Aspects of Biosemiotics Rainer E. Zimmermann IAG Philosophische Grundlagenprobleme, U Kassel / Clare Hall, UK – Cambridge / Lehrgebiet Philosophie, FB 13 AW, FH Muenchen, Lothstr. 34, D – 80335 München E-mail: [email protected] Abstract: According to recent work of Bounias and Bonaly cal terms with a view to the biosemiotic consequences. As this (2000), there is a close relationship between the conceptualiza- approach fits naturally into the Kassel programme of investigat- tion of biological life and mathematical conceptualization such ing the relationship between the cognitive perceiving of the that both of them co-depend on each other when discussing world and its communicative modeling (Zimmermann 2004a, preliminary conditions for properties of biosystems. More pre- 2005b), it is found that topology as formal nucleus of spatial cisely, such properties can be realized only, if the space of modeling is more than relevant for the understanding of repre- orbits of members of some topological space X by the set of senting and co-creating the world as it is cognitively perceived functions governing the interactions of these members is com- and communicated in its design. Also, its implications may well pact and complete. This result has important consequences for serve the theoretical (top-down) foundation of biosemiotics the maximization of complementarity in habitat occupation as itself. well as for the reciprocal contributions of sub(eco)systems with respect
    [Show full text]
  • Ph.D. University of Iowa 1983, Area in Geometric Topology Especially Knot Theory
    Ph.D. University of Iowa 1983, area in geometric topology especially knot theory. Faculty in the Department of Mathematics & Statistics, Saint Louis University. 1983 to 1987: Assistant Professor 1987 to 1994: Associate Professor 1994 to present: Professor 1. Evidence of teaching excellence Certificate for the nomination for best professor from Reiner Hall students, 2007. 2. Research papers 1. Non-algebraic killers of knot groups, Proceedings of the America Mathematical Society 95 (1985), 139-146. 2. Algebraic meridians of knot groups, Transactions of the American Mathematical Society 294 (1986), 733-747. 3. Isomorphisms and peripheral structure of knot groups, Mathematische Annalen 282 (1988), 343-348. 4. Seifert fibered surgery manifolds of composite knots, (with John Kalliongis) Proceedings of the American Mathematical Society 108 (1990), 1047-1053. 5. A note on incompressible surfaces in solid tori and in lens spaces, Proceedings of the International Conference on Knot Theory and Related Topics, Walter de Gruyter (1992), 213-229. 6. Incompressible surfaces in the knot manifolds of torus knots, Topology 33 (1994), 197- 201. 7. Topics in Classical Knot Theory, monograph written for talks given at the Institute of Mathematics, Academia Sinica, Taiwan, 1996. 8. Bracket and regular isotopy of singular link diagrams, preprint, 1998. 1 2 9. Regular isotopy of singular link diagrams, Proceedings of the American Mathematical Society 129 (2001), 2497-2502. 10. Normal holonomy and writhing number of polygonal knots, (with James Hebda), Pacific Journal of Mathematics 204, no. 1, 77 - 95, 2002. 11. Framing of knots satisfying differential relations, (with James Hebda), Transactions of the American Mathematics Society 356, no.
    [Show full text]
  • Arxiv:Math/0307077V4
    Table of Contents for the Handbook of Knot Theory William W. Menasco and Morwen B. Thistlethwaite, Editors (1) Colin Adams, Hyperbolic Knots (2) Joan S. Birman and Tara Brendle, Braids: A Survey (3) John Etnyre Legendrian and Transversal Knots (4) Greg Friedman, Knot Spinning (5) Jim Hoste, The Enumeration and Classification of Knots and Links (6) Louis Kauffman, Knot Diagramitics (7) Charles Livingston, A Survey of Classical Knot Concordance (8) Lee Rudolph, Knot Theory of Complex Plane Curves (9) Marty Scharlemann, Thin Position in the Theory of Classical Knots (10) Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory arXiv:math/0307077v4 [math.GT] 26 Nov 2004 A SURVEY OF CLASSICAL KNOT CONCORDANCE CHARLES LIVINGSTON In 1926 Artin [3] described the construction of certain knotted 2–spheres in R4. The intersection of each of these knots with the standard R3 ⊂ R4 is a nontrivial knot in R3. Thus a natural problem is to identify which knots can occur as such slices of knotted 2–spheres. Initially it seemed possible that every knot is such a slice knot and it wasn’t until the early 1960s that Murasugi [86] and Fox and Milnor [24, 25] succeeded at proving that some knots are not slice. Slice knots can be used to define an equivalence relation on the set of knots in S3: knots K and J are equivalent if K# − J is slice. With this equivalence the set of knots becomes a group, the concordance group of knots. Much progress has been made in studying slice knots and the concordance group, yet some of the most easily asked questions remain untouched.
    [Show full text]
  • Prospects in Topology
    Annals of Mathematics Studies Number 138 Prospects in Topology PROCEEDINGS OF A CONFERENCE IN HONOR OF WILLIAM BROWDER edited by Frank Quinn PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1995 Copyright © 1995 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 987654321 Library of Congress Cataloging-in-Publication Data Prospects in topology : proceedings of a conference in honor of W illiam Browder / Edited by Frank Quinn. p. cm. — (Annals of mathematics studies ; no. 138) Conference held Mar. 1994, at Princeton University. Includes bibliographical references. ISB N 0-691-02729-3 (alk. paper). — ISBN 0-691-02728-5 (pbk. : alk. paper) 1. Topology— Congresses. I. Browder, William. II. Quinn, F. (Frank), 1946- . III. Series. QA611.A1P76 1996 514— dc20 95-25751 The publisher would like to acknowledge the editor of this volume for providing the camera-ready copy from which this book was printed PROSPECTS IN TOPOLOGY F r a n k Q u in n , E d it o r Proceedings of a conference in honor of William Browder Princeton, March 1994 Contents Foreword..........................................................................................................vii Program of the conference ................................................................................ix Mathematical descendants of William Browder...............................................xi A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay........................................................................3 A.
    [Show full text]
  • On Finite Type 3-Manifold Invariants Iii: Manifold Weight Systems
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Pergamon Top&y?. Vol. 37, No. 2, PP. 221.243, 1998 ~2 1997 Elsevw Science Ltd Printed in Great Britain. All rights reserved 0040-9383/97 $19.00 + 0.00 PII: SOO40-9383(97)00028-l ON FINITE TYPE 3-MANIFOLD INVARIANTS III: MANIFOLD WEIGHT SYSTEMS STAVROSGAROUFALIDIS and TOMOTADAOHTSUKI (Received for publication 9 June 1997) The present paper is a continuation of [11,6] devoted to the study of finite type invariants of integral homology 3-spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3-spheres are determined (modulo invariants of type m - 1) by their associated manifold weight systems. In particular, we deduce a vanishing theorem for finite type invariants. We show that the space of manifold weight systems forms a commutative, co-commutative Hopf algebra and that the map from finite type invariants to manifold weight systems is an algebra map. We conclude with better bounds for the graded space of finite type invariants of integral homology 3-spheres. 0 1997 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION 1.1. History The present paper is a continuation of [ll, 63 devoted to the study of finite type invariants of oriented integral homology 3-spheres. There are two main sources of motivation for the present work: (perturbative) Chern-Simons theory in three dimensions, and Vassiliev invariants of knots in S3. Witten [16] in his seminal paper, using path integrals (an infinite dimensional “integra- tion” method) introduced a topological quantum field theory in three dimensions whose Lagrangian was the Chern-Simons function on the space of all connections.
    [Show full text]
  • The Hyperbolic Volume of Knots from Quantum Dilogarithm
    The hyperbolic volume of knots from quantum dilogarithm R.M. Kashaev∗ Laboratoire de Physique Th´eorique enslapp† ENSLyon, 46 All´ee d’Italie, 69007 Lyon, FRANCE E-mail: [email protected] January, 1996 Abstract The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particular examples it is argued that for a hyperbolic knot (link) the absolute value of this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate. arXiv:q-alg/9601025v2 25 Mar 1996 ENSLAPP-L-577/96 ∗On leave of absence from St. Petersburg Branch of the Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191011, RUSSIA †URA 14-36 du CNRS, associ´ee `al’E.N.S. de Lyon, au LAPP d’Annecy et `al’Universit`e de Savoie 1 1 Introduction Many known knot and link invariants, including Alexander [1] and Jones [2] polynomials, can be obtained from R-matrices, solutions to the Yang-Baxter equation (YBE) [3, 4]. Remarkably, many R-matrices in turn appear in quantum 3-dimensional Chern-Simons (CS) theory as partition functions of a 3-manifold with boundary, and with properly chosen Wilson lines [8]. The corresponding knot invariant acquires an interpretation in terms of a meanvalue of a Wilson loop. Note, that the quantum CS theory is an example of the topological quantum field theory (TQFT) defined axiomatically in [9]. Thurston in his theory of hyperbolic 3-manifolds [5] introduces the notion of a hyperbolic knot: a knot that has a complement that can be given a metric of negative constant curvature.
    [Show full text]
  • Exact Results for Perturbative Chern–Simons Theory with Complex Gauge Group Tudor Dimofte, Sergei Gukov, Jonatan Lenells and Don Zagier
    communications in number theory and physics Volume 3, Number 2, 363–443, 2009 Exact results for perturbative Chern–Simons theory with complex gauge group Tudor Dimofte, Sergei Gukov, Jonatan Lenells and Don Zagier We develop several methods that allow us to compute all-loop par- tition functions in perturbative Chern–Simons theory with com- plex gauge group GC, sometimes in multiple ways. In the back- ground of a non-abelian irreducible flat connection, perturbative GC invariants turn out to be interesting topological invariants, which are very different from the finite-type (Vassiliev) invariants usually studied in a theory with compact gauge group G and a triv- ial flat connection. We explore various aspects of these invariants and present an example where we compute them explicitly to high loop order. We also introduce a notion of “arithmetic topological quantum field theory” and conjecture (with supporting numerical evidence) that SL(2, C) Chern–Simons theory is an example of such a theory. 1. Introduction and summary 364 2. Perturbation theory around a non-trivial flat connection 373 2.1. Feynman diagrams and arithmetic TQFT 374 2.2. Quantization of Mflat(GC, Σ) 380 2.2.1 Analytic continuation. 384 2.2.2 A hierarchy of differential equations. 386 2.3. Classical and quantum symmetries 390 2.4. Brane quantization 394 3. A state integral model for perturbative SL(2, C) Chern−Simons theory 396 3.1. Ideal triangulations of hyperbolic 3-manifolds 397 363 364 Tudor Dimofte, Sergei Gukov, Jonatan Lenells and Don Zagier 3.2. Hikami’s invariant 402 3.3.
    [Show full text]
  • Vladimir Turaev, Friend and Colleague Athanase Papadopoulos
    Vladimir Turaev, friend and colleague Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Vladimir Turaev, friend and colleague. In: Topology and Geometry — A Collection of Essays Dedicated to Vladimir G. Turaev, European Mathematical Society Press, Berlin, p. 15-44, 2021, 978-3-98547-001-3. 10.4171/IRMA/33. hal-03285538 HAL Id: hal-03285538 https://hal.archives-ouvertes.fr/hal-03285538 Submitted on 13 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. VLADIMIR TURAEV, FRIEND AND COLLEAGUE ATHANASE PAPADOPOULOS Abstract. This is a biography and a report on the work of Vladimir Turaev. Using fundamental techniques that are rooted in classical to- pology, Turaev introduced new ideas and tools that transformed the field of knots and links and invariants of 3-manifolds. He is one of the main founders of the new topic called quantum topology. In surveying Turaev’s work, this article will give at the same time an overview of an important part of the intense activity in low-dimensional topology that took place over the last 45 years, with its connections with mathematical physics. The final version of this article appears in the book Topology and Geometry A Collection of Essays Dedicated to Vladimir G.
    [Show full text]
  • Refined BPS Invariants, Chern-Simons Theory, And
    Refined BPS Invariants, Chern-Simons Theory, and the Quantum Dilogarithm Thesis by Tudor Dan Dimofte In partial fulfillment of the requirements for the degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2010 (Defended March 30, 2010) ii c 2010 Tudor Dan Dimofte All Rights Reserved iii There are many individuals without whom the present thesis would not have been possible. Among them, I especially wish to thank Dan Jafferis, Sergei Gukov, Jonatan Lenells, Andy Neitzke, Hirosi Ooguri, Carol Silberstein, Yan Soibelman, Ketan Vyas, Masahito Yamazaki, Don Zagier, and Christian Zickert for a multitude of discussions and collaborations over innumerable hours leading to the culmination of this work. I am extremely grateful to Hirosi Ooguri and to Sergei Gukov, who have both advised me during various periods of my studies. I also wish to thank my family and friends — in particular Mom, Dad, and David — for their constant, essential support. iv Abstract In this thesis, we consider two main subjects: the refined BPS invariants of Calabi-Yau threefolds, and three-dimensional Chern-Simons theory with complex gauge group. We study the wall-crossing behavior of refined BPS invariants using a variety of techniques, including a four-dimensional supergravity analysis, statistical-mechanical melting crystal models, and relations to new mathematical invariants. We conjecture an equivalence be- tween refined invariants and the motivic Donaldson-Thomas invariants of Kontsevich and Soibelman. We then consider perturbative Chern-Simons theory with complex gauge group, combining traditional and novel approaches to the theory (including a new state integral model) to obtain exact results for perturbative partition functions.
    [Show full text]
  • CATEGORIFICATION in ALGEBRA and TOPOLOGY
    CATEGORIFICATION in ALGEBRA and TOPOLOGY List of Participants, Schedule and Abstracts of Talks Department of Mathematics Uppsala University . CATEGORIFICATION in ALGEBRA and TOPOLOGY Department of Mathematics Uppsala University Uppsala, Sweden September 6-11, 2006 Supported by • The Swedish Research Council Organizers: • Volodymyr Mazorchuk, Department of Mathematics, Uppsala University; • Oleg Viro, Department of Mathematics, Uppsala University. 1 List of Participants: 1. Ole Andersson, Uppsala University, Sweden 2. Benjamin Audoux, Universite Toulouse, France 3. Dror Bar-Natan, University of Toronto, Canada 4. Anna Beliakova, Universitt Zrich, Switzerland 5. Johan Bjorklund, Uppsala University, Sweden 6. Christian Blanchet, Universite Bretagne-Sud, Vannes, France 7. Jonathan Brundan, University of Oregon, USA 8. Emily Burgunder, Universite de Montpellier II, France 9. Paolo Casati, Universita di Milano, Italy 10. Sergei Chmutov, Ohio State University, USA 11. Alexandre Costa-Leite, University of Neuchatel, Switzerland 12. Ernst Dieterich, Uppsala University, Sweden 13. Roger Fenn, Sussex University, UK 14. Peter Fiebig, Freiburg University, Germany 15. Karl-Heinz Fieseler, Uppsala University, Sweden 16. Anders Frisk, Uppsala University, Sweden 17. Charles Frohman, University of Iowa, USA 18. Juergen Fuchs, Karlstads University, Sweden 19. Patrick Gilmer, Louisiana State University, USA 20. Nikolaj Glazunov, Institute of Cybernetics, Kyiv, Ukraine 21. Ian Grojnowski, Cambridge University, UK 22. Jonas Hartwig, Chalmers University of Technology, Gothenburg, Sweden 2 23. Magnus Hellgren, Uppsala University, Sweden 24. Martin Herschend, Uppsala University, Sweden 25. Magnus Jacobsson, INdAM, Rome, Italy 26. Zhongli Jiang, BGP Science, P.R. China 27. Louis Kauffman, University of Illinois at Chicago, USA 28. Johan K˚ahrstr¨om, Uppsala University, Sweden 29. Sefi Ladkani, The Hebrew University of Jerusalem, Israel 30.
    [Show full text]
  • The Jones Knot Polynomial Theorem Fora Knotor Link K, Withwrithew(K) and Kauffman Bracket K, the Jones Polynomial J(K) = (−X)3W(K)K Is a Knot Invariant
    THEOREM OF THE DAY The Jones Knot Polynomial Theorem Fora knotor link K, withwrithew(K) and Kauffman bracket K, the Jones polynomial J(K) = (−x)3w(K)K is a knot invariant. A knot invariant is a quantity which is invariant under deformation of a knot or link, without cutting, in three dimensions. Equivalently, we define the in- variance as under application of the three Reidemeister Moves to a 2D knot diagram representing the three-dimensional embedding schematically, in terms of over- and under-crossings. Our example concerns a 2-component link, the so-called ‘Solomon’s Seal’ knot (above) labelled, as an alternating 4-crossing link, L4a1. Each component is the ‘unknot’ whose diagram is a simple closed plane curve: . Now the Kauffman bracket for a collection of k separated un- knots ··· may be specified as ··· = (−x2 − x−2)k−1. But L4a1’s unknots are intertwined; this is resolved by a ‘smoothing’ operation: while ascending an over-crossing, , we switch to the under- crossing, either to the left, , a ‘+1 smoothing’ or the right, , a ‘−1 smoothing’ (the definition is rotated appropriately for non- vertical over-crossings). There is a corresponding definition of K as a recursive rule: = x + x−1 . A complete smoothing of an n-crossing knot K is thus a choice of one of 2n sequences of +’s and −’s. Let S be the collection of all such sequences. Each results in, say ki, separate unknots, and has a ‘positivity’, say pi, obtained by summing the ±1 values of its smoothings. Then the full definition of K is: pi 2 −2 ki−1 K = PS x (−x − x ) .
    [Show full text]
  • 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.
    [Show full text]