Knot Theory and Statistical Mechanics

1997 ◽  
Vol 11 (01n02) ◽  
pp. 39-49 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Statistical Mechanics ◽  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Scattering Amplitude ◽  
Basic Structure ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Quantum Link

This paper gives a self-contained exposition of the basic structure of quantum link invariants as state summations for a vacuum-vacuum scattering amplitude. Models of Vaughan Jones are expressed in this context. A simple proof is given that an important subset of these invariants are built from Vassiliev invariants of finite type.

scholarly journals HOMOMORPHIC EXPANSIONS FOR KNOTTED TRIVALENT GRAPHS

2013 ◽  
Vol 22 (01) ◽  
pp. 1250137 ◽  
Author(s):  
DROR BAR-NATAN ◽  
ZSUZSANNA DANCSO
Keyword(s):  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Large Set ◽  
Correction Factors ◽  
Connected Sums ◽  
Trivalent Graphs ◽  
Type Invariant ◽  
Knotted Trivalent Graphs ◽  
Standard Operations

It had been known since old times (works of Murakami–Ohtsuki, Cheptea–Le and the second author) that there exists a universal finite type invariant ("an expansion") Z old for knotted trivalent graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Z old under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear. In this paper we present two equivalent ways of modifying Z old into a new expansion Z, defined on "dotted knotted trivalent graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connected sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of KTGs retains all the good qualities that KTGs have — it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "algebraic knot theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (band-slide) move, first proven by Le, Murakami, Murakami and Ohtsuki.

scholarly journals On parabolic submonoids of a class of singular Artin monoids

2007 ◽  
Vol 82 (1) ◽  
pp. 29-37
Author(s):  
Noelle Antony
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Artin Groups ◽  
Parabolic Subgroups ◽  
Vassiliev Invariants ◽  
Sole Purpose

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

GAUSS CODES, QUANTUM GROUPS AND RIBBON HOPF ALGEBRAS

1993 ◽  
Vol 05 (04) ◽  
pp. 735-773 ◽  
Author(s):  
LOUIS H. KAUFFMAN
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Knot Theory ◽  
Link Invariants ◽  
Finite Dimensional ◽  
Quantum Link

By relating the diagrammatic foundations of knot theory with the structure of abstract tensors, quantum groups and ribbon Hopf algebras, specific expressions are derived for quantum link invariants. These expressions, when applied to the case of finite dimensional unimodular ribbon Hopf algebras, give rise to invariants of 3-manifolds.

On the skein theory of dichromatic links and invariants of finite type

2017 ◽  
Vol 26 (13) ◽  
pp. 1750092 ◽  
Author(s):  
Khaled Bataineh
Keyword(s):  
Finite Type ◽  
Partial Derivative ◽  
Polynomial Invariant ◽  
Partial Derivatives ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Combinatorial Formulas ◽  
Skein Theory

In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in [Formula: see text]. They introduced a multi-variable polynomial invariant [Formula: see text]. We use special substitutions for some of the parameters of the invariant [Formula: see text] to show how to deduce invariants of finite type from [Formula: see text] using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in [Formula: see text] We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

scholarly journals Rotational virtual knots and quantum link invariants

2015 ◽  
Vol 24 (13) ◽  
pp. 1541008 ◽  
Author(s):  
Louis H. Kauffman
Keyword(s):  
Knot Theory ◽  
Link Invariant ◽  
Quantum Invariants ◽  
Virtual Knot ◽  
Link Invariants ◽  
Virtual Knots ◽  
Virtual Knot Theory ◽  
Quantum Link ◽  
Knots And Links

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

VASSILIEV INVARIANTS AND DOUBLE DATING TANGLES

2002 ◽  
Vol 11 (04) ◽  
pp. 527-544 ◽  
Author(s):  
MYEONG-JU JEONG ◽  
CHAN-YOUNG PARK
Keyword(s):  
Finite Type ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Link Invariant ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Knot Invariant ◽  
Vassiliev Invariant ◽  
Framed Link

In [1], E. Appleboim introduced the notion of double dating linking class-P invariants of finite type for framed links with a fixed linking matrix P and showed that all Vassiliev link invariants are of finite type for any linking matrix and in [13], R. Trapp provided a necessary condition for a knot invariant to be a Vassiliev invariant by using twist sequences. In this paper we provide a necessary condition for a framed link invariant to be a DD-linking class-P invariant of finite type by considering sequence of links induced from a double dating tangle. As applications we give a generalization of R. Trapp's result to see whether a link invariant is a Vassiliev invariant or not and apply the criterion for all non-zero coefficients of the Jones, HOMFLY, Q-, and Alexander polynomial.

PL Link Isotopy, Essential Knotting and Quotients of Polynomials

1991 ◽  
Vol 34 (4) ◽  
pp. 536-541 ◽  
Author(s):  
Dale Rolfsen
Keyword(s):  
Knot Theory ◽  
Piecewise Linear ◽  
Rational Functions ◽  
Isotopy Class ◽  
Link Invariants ◽  
Link Theory ◽  
The Individual

AbstractPiecewise-linear (nonambient) isotopy of classical links may be regarded as link theory modulo knot theory. This note considers an adaptation of new (and old) polynomial link invariants to this theory, obtained simply by dividing a link's polynomial by the polynomials of the individual components. The resulting rational functions are effective in distinguishing isotopy classes of links, and in demonstrating that certain links are essentially knotted in the sense that every link in its isotopy class has a knotted component. We also establish geometric criteria for essential knotting of links.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

Cellular bases of generalised q-Schur algebras

2016 ◽  
Vol 162 (3) ◽  
pp. 533-560
Author(s):  
STEPHEN DOTY ◽  
ANTHONY GIAQUINTO
Keyword(s):  
Representation Theory ◽  
Finite Type ◽  
Basic Structure ◽  
Schur Algebras ◽  
Generators And Relations

AbstractStarting from their defining presentation by generators and relations, we develop the basic structure and representation theory of generalised q-Schur algebras of finite type.

TWIST SEQUENCES AND VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (03) ◽  
pp. 391-405 ◽  
Author(s):  
ROLLAND TRAPP
Keyword(s):  
Finite Type ◽  
Polynomial Growth ◽  
Difference Sequence ◽  
Uniform Limit ◽  
Topological Information ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Vassiliev Invariant

In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

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