New Skein Invariants of Links
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Coefficients of Homfly Polynomial and Kauffman Polynomial Are Not Finite Type Invariants
COEFFICIENTS OF HOMFLY POLYNOMIAL AND KAUFFMAN POLYNOMIAL ARE NOT FINITE TYPE INVARIANTS GYO TAEK JIN AND JUNG HOON LEE Abstract. We show that the integer-valued knot invariants appearing as the nontrivial coe±cients of the HOMFLY polynomial, the Kau®man polynomial and the Q-polynomial are not of ¯nite type. 1. Introduction A numerical knot invariant V can be extended to have values on singular knots via the recurrence relation V (K£) = V (K+) ¡ V (K¡) where K£, K+ and K¡ are singular knots which are identical outside a small ball in which they di®er as shown in Figure 1. V is said to be of ¯nite type or a ¯nite type invariant if there is an integer m such that V vanishes for all singular knots with more than m singular double points. If m is the smallest such integer, V is said to be an invariant of order m. q - - - ¡@- @- ¡- K£ K+ K¡ Figure 1 As the following proposition states, every nontrivial coe±cient of the Alexander- Conway polynomial is a ¯nite type invariant [1, 6]. Theorem 1 (Bar-Natan). Let K be a knot and let 2 4 2m rK (z) = 1 + a2(K)z + a4(K)z + ¢ ¢ ¢ + a2m(K)z + ¢ ¢ ¢ be the Alexander-Conway polynomial of K. Then a2m is a ¯nite type invariant of order 2m for any positive integer m. The coe±cients of the Taylor expansion of any quantum polynomial invariant of knots after a suitable change of variable are all ¯nite type invariants [2]. For the Jones polynomial we have Date: October 17, 2000 (561). -
Restrictions on Homflypt and Kauffman Polynomials Arising from Local Moves
RESTRICTIONS ON HOMFLYPT AND KAUFFMAN POLYNOMIALS ARISING FROM LOCAL MOVES SANDY GANZELL, MERCEDES GONZALEZ, CHLOE' MARCUM, NINA RYALLS, AND MARIEL SANTOS Abstract. We study the effects of certain local moves on Homflypt and Kauffman polynomials. We show that all Homflypt (or Kauffman) polynomials are equal in a certain nontrivial quotient of the Laurent polynomial ring. As a consequence, we discover some new properties of these invariants. 1. Introduction The Jones polynomial has been widely studied since its introduction in [5]. Divisibility criteria for the Jones polynomial were first observed by Jones 3 in [6], who proved that 1 − VK is a multiple of (1 − t)(1 − t ) for any knot K. The first author observed in [3] that when two links L1 and L2 differ by specific local moves (e.g., crossing changes, ∆-moves, 4-moves, forbidden moves), we can find a fixed polynomial P such that VL1 − VL2 is always a multiple of P . Additional results of this kind were studied in [11] (Cn-moves) and [1] (double-∆-moves). The present paper follows the ideas of [3] to establish divisibility criteria for the Homflypt and Kauffman polynomials. Specifically, we examine the effect of certain local moves on these invariants to show that the difference of Homflypt polynomials for n-component links is always a multiple of a4 − 2a2 + 1 − a2z2, and the difference of Kauffman polynomials for knots is always a multiple of (a2 + 1)(a2 + az + 1). In Section 2 we define our main terms and recall constructions of the Hom- flypt and Kauffman polynomials that are used throughout the paper. -
The Jones Polynomial Through Linear Algebra
The Jones polynomial through linear algebra Iain Moffatt University of South Alabama Workshop in Knot Theory Waterloo, 24th September 2011 I. Moffatt (South Alabama) UW 2011 1 / 39 What and why What we’ll see The construction of link invariants through R-matrices. (c.f. Reshetikhin-Turaev invariants, quantum invariants) Why this? Can do some serious math using material from Linear Algebra 1. Illustrates how math works in the wild: start with a problem you want to solve; figure out an easier problem that you can solve; build up from this to solve your original problem. See the interplay between algebra, combinatorics and topology! It’s my favourite bit of math! I. Moffatt (South Alabama) UW 2011 2 / 39 What we’re trying to do A knot is a circle, S1, sitting in 3-space R3. A link is a number of disjoint circles in 3-space R3. Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. I. Moffatt (South Alabama) UW 2011 3 / 39 What we’re trying to do A knot is a circle, S1, sitting in 3-space R3. A link is a number of disjoint circles in 3-space R3. Knots and links are considered up to isotopy. This means you can “move then round in space, but you can’t cut or glue them”. The fundamental problem in knot theory is to determine whether or not two links are isotopic. =? =? I. Moffatt (South Alabama) UW 2011 3 / 39 To do this we need knot invariants: F : Links (a set) such (Isotopy) ! that F(L) = F(L0) = L = L0, Aim: construct6 link invariants) 6 using linear algebra. -
Kontsevich's Integral for the Kauffman Polynomial Thang Tu Quoc Le and Jun Murakami
T.T.Q. Le and J. Murakami Nagoya Math. J. Vol. 142 (1996), 39-65 KONTSEVICH'S INTEGRAL FOR THE KAUFFMAN POLYNOMIAL THANG TU QUOC LE AND JUN MURAKAMI 1. Introduction Kontsevich's integral is a knot invariant which contains in itself all knot in- variants of finite type, or Vassiliev's invariants. The value of this integral lies in an algebra d0, spanned by chord diagrams, subject to relations corresponding to the flatness of the Knizhnik-Zamolodchikov equation, or the so called infinitesimal pure braid relations [11]. For a Lie algebra g with a bilinear invariant form and a representation p : g —» End(V) one can associate a linear mapping WQtβ from dQ to C[[Λ]], called the weight system of g, t, p. Here t is the invariant element in g ® g corresponding to the bilinear form, i.e. t = Σa ea ® ea where iea} is an orthonormal basis of g with respect to the bilinear form. Combining with Kontsevich's integral we get a knot invariant with values in C [[/*]]. The coefficient of h is a Vassiliev invariant of degree n. On the other hand, for a simple Lie algebra g, t ^ g ® g as above, and a fi- nite dimensional irreducible representation p, there is another knot invariant, con- structed from the quantum i?-matrix corresponding to g, t, p. Here i?-matrix is the image of the universal quantum i?-matrix lying in °Uq(φ ^^(g) through the representation p. The construction is given, for example, in [17, 18, 20]. The in- variant is a Laurent polynomial in q, by putting q — exp(h) we get a formal series in h. -
Coloring Link Diagrams and Conway-Type Polynomial of Braids
COLORING LINK DIAGRAMS AND CONWAY-TYPE POLYNOMIAL OF BRAIDS MICHAEL BRANDENBURSKY Abstract. In this paper we define and present a simple combinato- rial formula for a 3-variable Laurent polynomial invariant I(a; z; t) of conjugacy classes in Artin braid group Bm. We show that the Laurent polynomial I(a; z; t) satisfies the Conway skein relation and the coef- −k ficients of the 1-variable polynomial t I(a; z; t)ja=1;t=0 are Vassiliev invariants of braids. 1. Introduction. In this work we consider link invariants arising from the Alexander-Conway and HOMFLY-PT polynomials. The HOMFLY-PT polynomial P (L) is an invariant of an oriented link L (see for example [10], [18], [23]). It is a Laurent polynomial in two variables a and z, which satisfies the following skein relation: ( ) ( ) ( ) (1) aP − a−1P = zP : + − 0 The HOMFLY-PT polynomial is normalized( in) the following way. If Or is r−1 a−a−1 the r-component unlink, then P (Or) = z . The Conway polyno- mial r may be defined as r(L) := P (L)ja=1. This polynomial is a renor- malized version of the Alexander polynomial (see for example [9], [17]). All coefficients of r are finite type or Vassiliev invariants. Recently, invariants of conjugacy classes of braids received a considerable attention, since in some cases they define quasi-morphisms on braid groups and induce quasi-morphisms on certain groups of diffeomorphisms of smooth manifolds, see for example [3, 6, 7, 8, 11, 12, 14, 15, 19, 20]. In this paper we present a certain combinatorial construction of a 3-variable Laurent polynomial invariant I(a; z; t) of conjugacy classes in Artin braid group Bm. -
The Kauffman Bracket and Genus of Alternating Links
California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 6-2016 The Kauffman Bracket and Genus of Alternating Links Bryan M. Nguyen Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Other Mathematics Commons Recommended Citation Nguyen, Bryan M., "The Kauffman Bracket and Genus of Alternating Links" (2016). Electronic Theses, Projects, and Dissertations. 360. https://scholarworks.lib.csusb.edu/etd/360 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. The Kauffman Bracket and Genus of Alternating Links A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Bryan Minh Nhut Nguyen June 2016 The Kauffman Bracket and Genus of Alternating Links A Thesis Presented to the Faculty of California State University, San Bernardino by Bryan Minh Nhut Nguyen June 2016 Approved by: Dr. Rolland Trapp, Committee Chair Date Dr. Gary Griffing, Committee Member Dr. Jeremy Aikin, Committee Member Dr. Charles Stanton, Chair, Dr. Corey Dunn Department of Mathematics Graduate Coordinator, Department of Mathematics iii Abstract Giving a knot, there are three rules to help us finding the Kauffman bracket polynomial. Choosing knot's orientation, then applying the Seifert algorithm to find the Euler characteristic and genus of its surface. Finally finding the relationship of the Kauffman bracket polynomial and the genus of the alternating links is the main goal of this paper. -
Lecture 1 Knots and Links, Reidemeister Moves, Alexander–Conway Polynomial
Lecture 1 Knots and links, Reidemeister moves, Alexander–Conway polynomial 1.1. Main definitions A(nonoriented) knot is defined as a closed broken line without self-intersections in Euclidean space R3.A(nonoriented) link is a set of pairwise nonintertsecting closed broken lines without self-intersections in R3; the set may be empty or consist of one element, so that a knot is a particular case of a link. An oriented knot or link is a knot or link supplied with an orientation of its line(s), i.e., with a chosen direction for going around its line(s). Some well known knots and links are shown in the figure below. (The lines representing the knots and links appear to be smooth curves rather than broken lines – this is because the edges of the broken lines and the angle between successive edges are tiny and not distinguished by the human eye :-). (a) (b) (c) (d) (e) (f) (g) Figure 1.1. Examples of knots and links The knot (a) is the right trefoil, (b), the left trefoil (it is the mirror image of (a)), (c) is the eight knot), (d) is the granny knot; 1 2 the link (e) is called the Hopf link, (f) is the Whitehead link, and (g) is known as the Borromeo rings. Two knots (or links) K, K0 are called equivalent) if there exists a finite sequence of ∆-moves taking K to K0, a ∆-move being one of the transformations shown in Figure 1.2; note that such a transformation may be performed only if triangle ABC does not intersect any other part of the line(s). -
Polynomial Invariants and Vassiliev Invariants 1 Introduction
ISSN 1464-8997 (on line) 1464-8989 (printed) 89 eometry & opology onographs G T M Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001) Pages 89–101 Polynomial invariants and Vassiliev invariants Myeong-Ju Jeong Chan-Young Park Abstract We give a criterion to detect whether the derivatives of the HOMFLY polynomial at a point is a Vassiliev invariant or not. In partic- (m,n) ular, for a complex number b we show that the derivative PK (b, 0) = ∂m ∂n ∂am ∂xn PK (a, x) (a,x)=(b,0) of the HOMFLY polynomial of a knot K at (b, 0) is a Vassiliev| invariant if and only if b = 1. Also we analyze the ± space Vn of Vassiliev invariants of degree n for n =1, 2, 3, 4, 5 by using the ¯–operation and the ∗ –operation in [5].≤ These two operations are uni- fied to the ˆ –operation. For each Vassiliev invariant v of degree n,v ˆ is a Vassiliev invariant of degree n and the valuev ˆ(K) of a knot≤ K is a polynomial with multi–variables≤ of degree n and we give some questions on polynomial invariants and the Vassiliev≤ invariants. AMS Classification 57M25 Keywords Knots, Vassiliev invariants, double dating tangles, knot poly- nomials 1 Introduction In 1990, V. A. Vassiliev introduced the concept of a finite type invariant of knots, called Vassiliev invariants [13]. There are some analogies between Vassiliev invariants and polynomials. For example, in 1996 D. Bar–Natan showed that when a Vassiliev invariant of degree m is evaluated on a knot diagram having n crossings, the result is approximately bounded by a constant times of nm [2] and S. -
An Introduction to Knot Theory
An Introduction to Knot Theory Matt Skerritt (c9903032) June 27, 2003 1 Introduction A knot, mathematically speaking, is a closed curve sitting in three dimensional space that does not intersect itself. Intuitively if we were to take a piece of string, cord, or the like, tie a knot in it and then glue the loose ends together, we would have a knot. It should be impossible to untangle the knot without cutting the string somewhere. The use of the word `should' here is quite deliberate, however. It is possible, if we have not tangled the string very well, that we will be able to untangle the mess we created and end up with just a circle of string. Of course, this circle of string still ¯ts the de¯nition of a closed curve sitting in three dimensional space and so is still a knot, but it's not very interesting. We call such a knot the trivial knot or the unknot. Knots come in many shapes and sizes from small and simple like the unknot through to large and tangled and messy, and beyond (and everything in between). The biggest questions to a knot theorist are \are these two knots the same or di®erent" or even more importantly \is there an easy way to tell if two knots are the same or di®erent". This is the heart of knot theory. Merely looking at two tangled messes is almost never su±cient to tell them apart, at least not in any interesting cases. Consider the following four images for example. -
On the Study of Chirality of Knots Through Polynomial Invariants
Treball final de grau GRAU DE MATEMÀTIQUES Facultat de Matemàtiques i Informàtica Universitat de Barcelona KNOT THEORY: On the study of chirality of knots through polynomial invariants Autor: Sergi Justiniano Claramunt Director: Dr. Javier José Gutiérrez Marín Realitzat a: Departament de Matemàtiques i Informàtica Barcelona, January 18, 2019 Contents Abstract ii Introduction iii 1 Mathematical bases 1 1.1 Definition of a knot . .1 1.2 Equivalence of knots . .4 1.3 Knot projections and diagrams . .6 1.4 Reidemeister moves . .8 1.5 Invariants . .9 1.6 Symmetries, properties and generation of knots . 11 1.7 Tangles and Conway notation . 12 2 Jones Polynomial 15 2.1 Introduction . 15 2.2 Rules of bracket polynomial . 16 2.3 Writhe and invariance of Jones polynomial . 18 2.4 Main theorems and applications . 22 3 HOMFLY and Kauffman polynomials on chirality detection 25 3.1 HOMFLY polynomial . 25 3.2 Kauffman polynomial . 28 3.3 Testing chirality . 31 4 Conclusions 33 Bibliography 35 i Abstract In this project we introduce the theory of knots and specialize in the compu- tation of the knot polynomials. After presenting the Jones polynomial, its two two-variable generalizations are also introduced: the Kauffman and HOMFLY polynomial. Then we study the ability of these polynomials on detecting chirality, obtaining a knot not detected chiral by the HOMFLY polynomial, but detected chiral by the Kauffman polynomial. Introduction The main idea of this project is to give a clear and short introduction to the theory of knots and in particular the utility of knot polynomials on detecting chirality of knots. -
Kauffman's Polynomial and Alternating Links
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector 0040 9383 8X S3M+ 64 t 1988 Pergamon Press plc KAUFFMAN’S POLYNOMIAL AND ALTERNATING LINKS MORWEN B. THISTLETHWAITE (Received 17 November 1986) IN THIS article, a connection is established between certain terms of the Kauffman poly- nomial F,(a, z) of a non-split, alternating link L, and certain terms of the Tutte polynomial x&x, y) of the graph G associated with a black-and-white colouring of the regions of a reduced alternating diagram of L. This has the following geometric consequences: THEOREM 1. The writhe of a reduced alternating diagram of an alternating link L is an isotopy invariant of L. THEOREM 2. If a prime link L has a reduced alternating diagram with Conway basic polyhedron l*, 6* or 8*, then any other reduced alternating diagram of L has the same Conway basic polyhedron. These theorems result in quick tests which will often distinguish between links having alternating diagrams with the same number of crossings; links having reduced alternating diagrams with different numbers of crossings are automatically distinguished by Cor- ollary 1 of [12] (proved also in [4] and [9]). Also, as L. H. Kauffman has pointed out, it follows from Theorem 1 that any reduced alternating diagram of an amphicheiral alterna- ting knot must have writhe 0. K. Murasugi, in [lo], has independently obtained a different proof of Theorem 1; he has discovered a formula relating the extreme powers of t in the Jones polynomial VL(t) with the writhe of the reduced alternating diagram of L and the signature of L. -
Knotting Mathematics and Art : International Conference in Low Dimensional Topology and Mathematical Art
Knotting Mathematics and Art : International Conference in Low Dimensional Topology and Mathematical Art November 1 – 4, 2007 University of South Florida, Tampa, Florida, U.S.A. Abstract (Nagle Lecture, Opening Talk) John H. Conway (Princeton University) From Topology to Symmetry (This talk will be the Nagle Lecture, http://www.math.usf.edu/Nagle/index.html, scheduled on November 1, 7:30–8:30PM, also regarded as an opening lecture of the conference.) It is well known that the symmetries of repeating patterns in the Euclidean plane belong to just one of 17 groups. Less well-known are the corresponding enumerations for frieze patterns (7 types) and patterns on the sphere (7 particular groups + 7 infinite series) and the hyperbolic plane, which supports a multiple infinity of groups. I shall describe some of these, and perhaps say something about the corresponding enumerations in 3 and more dimensions. Abstract (Plenary Talks) (In alphabetical order of the author’s last name) Thomas Banchoff (Brown University) The Fourth Dimension and Salvador Dali Salvador Dali was fascinated by geometric objects in the fourth dimension, including hyper- cubes and catastrophe surfaces. Where did he get his inspirations and how did he represent four- dimensional phenomena in his paintings? This talk will describe contacts with the artist over a ten-year period starting in 1975 as well as some new information about his sources. J. Scott Carter (University of South Alabama), Tony Robbin (Artist and author) Collaboration across the the two cultures Starting from a mathematically simple idea — that the sum of cubes can be written as the square of a triangle — we begin investigating the aesthetics of the product of two isosoles right triangles from our respective disciplines.