New Developments in the Theory of Knots

The following sections are included:

• Preface

• CONTENTS

A theorem of J. Alexander [1] asserts that any tame oriented link in 3-space may be represented by a pair (b, n), where b is an element of the n-string braid group B_n. The link L is obtained by closing b, i.e., tying the top end of each string to the same position on the bottom of the braid as shown in Figure 1. The closed braid will be denoted 6^{^}…

The following sections are included:

• THE FREYD-YETTER APPROACH

• THE HOSTE APPROACH

• THE LICKORISH AND MILLETT APPROACH

• THE OCNEANU APPROACH

• REFERENCES

By studying representations of the braid group satisfying a certain quadratic relation we obtain a polynomial invariant in two variables for oriented links. It is expressed using a trace, discovered by Ocneanu, on the Hecke algebras of type A. A certain specialization of the polynomial, whose discovery predated and inspired the two-variable one, is seen to come in two inequivalent ways, from a Hecke algebra quotient and a linear functional on it which has already been used in statistical mechanics. The two-variable polynomial was first discovered by Freyd-Yetter, Lickorish-Millet, Ocneanu, Hoste, and Przytycki-Traczyk.

A new invariant polynomial for knots and links is constructed from a solvable vertex model describing a critical statistical system. Various implications and the possible generalizations are discussed in connection with the recent development in the study of critical phenomena in two dimensions.

Presented is a general method to construct representations of the braid group, a basic object in the knot theory, from the Boltzmann weights of the exactly solvable models in statistical mechanics at criticality. The method is applied to a class of N-state vertex models (N=2, 3 and 4) to have explicit braid group representations. Furthermore, the Markov traces are introduced and a sequence of new link polynomials, topological invariants for knots and links, is constructed.

Using a generalized Alexander-Conway relation derived from a three-state exactly solvable model in statistical mechanics, new invariant polynomials for knots and links are explicitly evaluated. It is shown that the invariant polynomials for closed 3-braids are obtained recursively. It is also shown that the invariant polynomials are more powerful than the Jones polynomials.

New link polynomials, reported in I and II of the series, are extended into those with two variables. A concept of composite string is introduced. It is shown that the composite string representation and the generalized Ocneanu's trace lead to a sequence of two-variable link polynomials. In addition, algebraic aspects of the composite string representation are studied in some detail.

We present a general method to construct link polynomials, invariants for knots and links, from the exactly solvable IRF (Interaction Round a Face) models in statistical mechanics which satisfy the Yang-Baxter relation.

Yang-Baxter operator is shown to be a fundamental object which relates theory of solvable models to theory of knots and links. First, general properties of Yang-Baxter operators are investigated. Second, a method to construct composite Yang-Baxter operators is explicitly shown. Lastly, from Yang-Baxter operators with crossing symmetry, braid-monoid algebras are derived. It is emphasized that the factorized S-matrices and their graphical illustrations link two approaches, algebraic and combinatorial, in the knot theory.

The following sections are included:

• INTRODUCTION

• BRACKET INVARIANT

• CHIRALITY OF ALTERNATING LINKS

• BRAID STATES AND THE DIAGRAM ALGEBRA

• REFERENCES

The following sections are included:

• Introduction

• The Yang-Baxter operators

• Invariants of braids and links

• Examples and applications

• State models for the invariants of links

• Proof of Theorem 4.3.4

• References

The following sections are included:

• INTRODUCTION

• Algebra U_q(sl(2))

• Irreducible Representations of U_q(su(2)) and Finite Dimensional R-Matrices

• q-Analog of Clebsch-Gordan Coefficients

• Graphical Representation of R-matrices And q-Clebsch-Gordan Coefficients

• The q-Analogs Of The Wigner-Rakah 6j-Symbols

• Graphical Representaion of 9-6j-Symbols

• The Invariants Of Links Associated With U_q(su(2))

• CONCLUSION

• References

We present a general method to construct the sequence of new link polynomials and its two variable extension from exactly solvable models in statistical mechanics. First, we find representations of the braid group from the Boltzmann weights of the exactly solvable models. Second, we give the Markov traces associated with new braid group representations and using them construct new link polynomials. Third, we extend the theory into a two-variable version of the new link polynomials. Throughout the paper, we emphasize the essential roles played by the exactly solvable models and the underlying Yang-Baxter relation.

It is shown that, to each finite dimensional representation of a quantized universal enveloping algebra, one can associate a vertex model as defined by Vaughan Jones in his constructions of isotopy link invariants.

This paper studies a relationship between the formalism of knot theory and certain models in statistical mechanics. It is shown how the partition function for the Potts model may be computed from an associated link diagram, and how this provides a common algorithmic model with the Jones polynomial. Certain features of both Jones polynomial and the Potts model can be treated in common, such as the appearance of the Temperley–Lieb algebra for braid diagrams and the geometry of the ice–model for piecewise linear diagrams.

We use three different kinds of statistical mechanical models to construct link invariants. The vertex models emerge as the most general. Our treatment of them is essentially the same as Turaev's. Using the work of Goldschmidt we are able to define models whose invariants are homology invariants for branched covers. Thus the statistical mechanical framework embraces both the “classical” and the “new” link invariants.

The following sections are included:

• Introduction

• Acknowledgement

• The parallel version of link invariants, I (the case of knots)

• The parallel version of link invariants, II (the case of general links)

• The r-parallel version of the two-variable Jones polynomial

• The r-parallel version of the one-variable jones polynomial

• The parallel version of the Kauffman polynomial

• Mutation and the r-parallel version of a link invariant

• References

The following sections are included:

• Introduction

• Generalities
  • The Global Index
  • The Local Index
  • Examples

• The Basic Construction
  • Extending Finite von Neumann Algebras by Subalgebras
  • Inclusions of Complex Semisimple Algebras
  • Finite Dimensional C^*-Algebras
  • Two Extensions; Goldman's Theorem

• Possible Values of the Index
  • Certain Algebras Generated by Projections
  • Restrictions on τ
  • Values of the Index

• The Bratteli Diagram; the Relative Commutant
  • The Bratteli Diagram when τ ≦ 1/4
  • The Bratteli Diagrams
  • The Relative Commutant when τ ≦ 1/4

• References

The following sections are included:

• Construction of subfactors by AF algebras

• Orthogonal representations of Hecke algebras of type A_n

• Markov traces for Hecke algebras

• References

A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on 2 parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras.

The following sections are included:

• Introduction

• The Kauffman polynomial and the L-polynomial

• Rectangular diagrams

• Basic deformations

• Knit semi-groups

• An algebra associated with the L-polynomial

• A candidate Ē₃(α, β) for E₃(α, β)

• Structure of Ē₃(α, β)

• Irreducible representations of D₃ through Ē₃(α, β)

• Formulas for the L-polynomial, the Kauffman polynomial and the Q-polynomial of a closed 3-knit

• E₃(α, β) is isomorphic to Ē₃(α, β)

• The Kauffman polynomials of closed 3-braids

• The Q-polynomial of closed 3-braids

• Examples for Theorem 12.2

• The Kauffman polynomials of 2-bridge links

• Examples for Theorem 15.1

• References

The following sections are included:

• INTRODUCTION

• AXIOMS AND COMPUTATIONS

• THE MAIN THEOREM

• APPLICATIONS

• REFERENCES

The following sections are included:

• Introduction

• Notations and definitions

• Bunching operations

• Lemma and Theorem 1

• Applications

• Acknowledgement

• References

The following sections are included:

• Introduction

• The Jones polynomial

• The Alexander polynomial

• The examples

• A conjecture

• References

Let J_K(t) = a_rt^r + ⋯ + a_st^s, r > s, be the Jones polynomial of a knot K in S³. For an alternating knot, it is proved that r − s is bounded by the number of double points in any alternating projection of K. This upper bound is attained by many alternating knots, including 2-bridge knots, and therefore, for these knots, r − s gives the minimum number of double points among all alternating projections of K. If K is a special alternating knot, it is also proved that a_s = 1 and s is equal to the genus of K. Similar results hold for links.

The following sections are included:

• INTRODUCTION AND MAIN THEOREMS

• PROOF OF THEOREM 1

• PROOFS OF THEOREMS 2–4

• REFERENCES

The following sections are included:

• Introduction

• The proof of the theorem

• Basic properties of the polynomials

• References

The 2-variable polynomial P_K of a satellite K is shown not to satisfy any formula, relating it to the polynomial of its companion and of the pattern, which is at all similar to the formulae for Alexander polynomials. Examples are given of various pairs of knots which can be distinguished by calculating P for 2-strand cables about them even though the knots themselves share the same P. Properties of a given knot such as braid index and amphicheirality, which may not be apparent from the knot's polynomial P, are shown in certain cases to be detectable from the polynomial of a 2-cable about the knot.

In [1] V. F. R. Jones introduces a polynomial invariant V_L(t) for an oriented link L and there he states that d/dtV_K(1)=0 for any knot K. In this note we generalize this to links. Moreover we give a formula which expresses d²/dt²V_L(1) in terms of the z²-terms of the Conway polynomials [2, § 5] and linking numbers of the components of L…

In [8] V. F. R. Jones introduced a Laurent polynomial invariant V_L(t) for an oriented link L in S³, and there he gave formulae which relate V_L(t) and the Alexander polynomial for three- or four-braid knots. The Jones polynomial V_L(t) has been generalized to the two-variable polynomial P_L(a, z) by several authors [4,6,12,14]. In this paper we give a formula for P_L(a, z) which states that for a link with n Seifert circles P_L(a₀, z), P_L(a₁, z), …, and P_L(a_n, z) are “linearly dependent”, where a₀, a₁, …, and a_n are mutually independent variables. This formula can be specialized to the formulae in [8]…

The following sections are included:

• INTRODUCTION

• THE EXTENDED DUAL STATE LEMMA

• KAUFFMAN'S STATE MODEL FOR THE JONES POLYNOMIAL

• PROOF OF THEOREM 1

• PROOF OF THEOREM 2

• APPENDIX: AN IMPROVEMENT OF THE INEQUALITY OF THEOREM 1

• REFERENCES

We prove that the (n,k)-cables around a mutant pair of knots C₁ and C₂ cannot be distinguished by the Jones polynomial V, for any (n,k). we prove further that the same result holds for any other satellite around C₁ and C₂.

A link L_β(2k, n – 2k) is defined by a type (2k, n – 2k) pairing of an n-braid β if the first 2k strands are joined up as in a plat and the remaining n – 2k as in a closed braid. The main result is a formula for the Jones polynomials of L_β(2k, n – 2k), valid for all k, 0 ≤ 2k ≤ n, which generalizes and relates earlier results of Jones for the cases n = 0 and 2k.

The following sections are included:

• Introduction

• Affine lie Algebra of type
  • Lie Algebra of type A₁ and its finite-dimensional modules
  • The affine Lie algebra of type
  • Segal-Sugawara form

• Vertex Operators (Primary fields)
  • Field operators
  • Vertex operators
  • Proof of Proposition 2.1 and Theorem 2.3
  • Normalization of vertex operators and Proof of Lemma 2.2.
  • Operator product expansions
  • Actions of ĝ and ℒ on vertex operators

• Differential Equations of N-point Functions and Composability of Vertex Operators
  • N-point functions and their differential equations
  • Solutions of fundamental equation
  • Composability of vertex operators

• Commutation Relations of Vertex Operators
  • Formulation of the problem
  • Reduced Equation
  • Connection matrices for J = (j₄, ½, j₂, j₁)
  • Case J = (j₄, ½, ½, j₁)

• Monodromy Representations of Braid Groups
  • Braid groups and Hecke algebras
  • Monodromy representations
  • Wenzl's representations of Hecke algebra
  • Fusion rule

• Appendix I. Bases of Tensor Products of sl-modules

• Appendix II. Connection Matrices of Reduced Equation
  • Solutions of reduced equation
  • Connection matrices of reduced equation

• References

We give corrections or comments to the five points in our paper [1].

The following sections are included:

• INTRODUCTION

• MONODROMY OF INTEGRABLE CONNECTIONS ARISING FROM CLASSICAL YANG-BAXTER EQUATIONS
  • Construction of connections
  • Description of the monodromy by means of solutions of quantum Yang-Baxter equations
  • Proof of Theorem 1.2.8.

• MONODROMY OF n-POINT FUNCTIONS IN TWO DIMENSIONAL CONFORMAL FIELD THEORY
  • Review of model due to Tsuchiya and Kanie
  • Monodromy associated with higher representations of sl(2,C)
  • Unitarity of the monodromy of n-point functions

• BIBLIOGRAPHY

The following sections are included:

• Introduction

• Dual algebras in two-dimensional quantum field models

• Statistics of “local fields” in two-dimensions, and the Yang-Baxter equation

• Statistics matrices, representations of braid groups and monodromy of Euclidean Green functions

• The example of the Wess-Zumino-Witten models

• Connections to the theory of knots and links

• Notes and open problems

• REFERENCES

Within the restricted context of conformal QFTh₂ we present a systematic analysis of the exchange algebras of light-cone fields which result from the previously studied global decomposition theory of Einstein-causal fields. Although certain aspects of the representation theory of exchange algebras with their Artin braid structure-constants appear in our illustrative examples (minimal and WZW models), our main interests are algebraic aspects. We view the present work as a new non-lagrangian (non-hamiltonian) approach to non-perturbative QFTh.

It is shown that 2 + 1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized from S³ to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1 + 1 dimensions.

The following sections are included:

• Jones and 2-Variable Jones Polynomials

• Q Polynomial

• Alexander Polynomial

• K(a,b)

• K(p₁,p₂, ⋯, p_n), n ≧ 3

• 2-Bridge Knots and Links

• References

The following sections are included:

• References