Greg Mcnulty March 17, 2005
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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). -
Section 10.2. New Polynomial Invariants
10.2. New Polynomial Invariants 1 Section 10.2. New Polynomial Invariants Note. In this section we mention the Jones polynomial and give a recursion for- mula for the HOMFLY polynomial. We give an example of the computation of a HOMFLY polynomial. Note. Vaughan F. R. Jones introduced a new knot polynomial in: A Polynomial Invariant for Knots via von Neumann Algebra, Bulletin of the American Mathe- matical Society, 12, 103–111 (1985). A copy is available online at the AMS.org website (accessed 4/11/2021). This is now known as the Jones polynomial. It is also a knot invariant and so can be used to potentially distinguish between knots. Shortly after the appearance of Jones’ paper, it was observed that the recursion techniques used to find the Jones polynomial and Conway polynomial can be gen- eralized to a 2-variable polynomial that contains information beyond that of the Jones and Conway polynomials. Note. The 2-variable polynomial was presented in: P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, and A. Ocneanu, A New Polynomial of Knots and Links, Bul- letin of the American Mathematical Society, 12(2): 239–246 (1985). This paper is also available online at the AMS.org website (accessed 4/11/2021). Based on a permutation of the first letters of the names of the authors, this has become knows as the “HOMFLY polynomial.” 10.2. New Polynomial Invariants 2 Definition. The HOMFLY polynomial, PL(`, m), is given by the recursion formula −1 `PL+ (`, m) + ` PL− (`, m) = −mPLS (`, m), and the condition PU (`, m) = 1 for U the unknot. -
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. -
Rotationally Symmetric Rose Links
Rose-Hulman Undergraduate Mathematics Journal Volume 14 Issue 1 Article 3 Rotationally Symmetric Rose Links Amelia Brown Simpson College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Brown, Amelia (2013) "Rotationally Symmetric Rose Links," Rose-Hulman Undergraduate Mathematics Journal: Vol. 14 : Iss. 1 , Article 3. Available at: https://scholar.rose-hulman.edu/rhumj/vol14/iss1/3 Rose- Hulman Undergraduate Mathematics Journal Rotationally Symmetric Rose Links Amelia Browna Volume 14, No. 1, Spring 2013 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 Email: [email protected] a http://www.rose-hulman.edu/mathjournal Simpson College Rose-Hulman Undergraduate Mathematics Journal Volume 14, No. 1, Spring 2013 Rotationally Symmetric Rose Links Amelia Brown Abstract. This paper is an introduction to rose links and some of their properties. We used a series of invariants to distinguish some rose links that are rotationally symmetric. We were able to distinguish all 3-component rose links and narrow the bounds on possible distinct 4 and 5-component rose links to between 2 and 8, and 2 and 16, respectively. An algorithm for drawing rose links and a table of rose links with up to five components are included. Acknowledgements: The initial cursory observations on rose links were made during the 2011 Dr. Albert H. and Greta A. Bryan Summer Research Program at Simpson College in conjunction with fellow students Michael Comer and Jes Toyne, with Dr. William Schellhorn serving as research advisor. The research presented in this paper, including developing the definitions and related concepts about rose links and applying invariants, was conducted as a part of the author's senior research project at Simpson College, again under the advisement of Dr. -
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). -
Knot Polynomials
Knot Polynomials André Schulze & Nasim Rahaman July 24, 2014 1 Why Polynomials? First introduced by James Wadell Alexander II in 1923, knot polynomials have proved themselves by being one of the most efficient ways of classifying knots. In this spirit, one expects two different projections of a knot to have the same knot polynomial; one therefore demands that a good knot polynomial be invariant under the three Reide- meister moves (although this is not always case, as we shall find out). In this report, we present 5 selected knot polynomials: the Bracket, Kauffman X, Jones, Alexander and HOMFLY polynomials. 2 The Bracket Polynomial 2.1 Calculating the Bracket Polynomial The Bracket polynomial makes for a great starting point in constructing knotpoly- nomials. We start with three simple rules, which are then iteratively applied to all crossings in the knot: A direct application of the third rule leads to the following relation for (untangled) unknots: The process of obtaining the Bracket polynomial can be streamlined by evaluating the contribution of a particular sequence of actions in undoing the knot (states) and summing over all such contributions to obtain the net polynomial. 2.2 The Problem with Bracket Polynomials The bracket polynomials can be shown to be invariant under types 2 and 3Reidemeis- ter moves. However by considering type 1 moves, its one major drawback becomes apparent. From: 1 we conclude that that the Bracket polynomial does not remain invariant under type 1 moves. This can be fixed by introducing the writhe of a knot, as we shallsee in the next section. -
New Skein Invariants of Links
NEW SKEIN INVARIANTS OF LINKS LOUIS H. KAUFFMAN AND SOFIA LAMBROPOULOU Abstract. We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, H[R], K[Q] and D[T ], based on the invariants of knots, R, Q and T , denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well- definedness of these invariants. These invariants are also reformulated into summations of the generating invariants (R, Q, T ) on sublinks of a given link L, obtained by partitioning L into collections of sublinks. Contents Introduction1 1. Previous work9 2. Skein theory for the generalized invariants 11 3. Closed combinatorial formulae for the generalized invariants 28 4. Conclusions 38 5. Discussing mathematical directions and applications 38 References 40 Introduction Skein theory was introduced by John Horton Conway in his remarkable paper [16] and in numerous conversations and lectures that Conway gave in the wake of publishing this paper. He not only discovered a remarkable normalized recursive method to compute the classical Alexander polynomial, but he formulated a generalized invariant of knots and links that is called skein theory. -
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.