Braid Group, Knot Theory and Statistical Mechanics

The following sections are included:

• FOREWORD

• CONTENTS

The following sections are included:

• Introduction

• Spin Models in Statistical Mechanics

• von Neumann Algebras

• Subfactors

• The Tower of Relative Commutants

• Some Examples of Bratteli Diagrams
  • The Q-state Potts model
  • The Fateev-Zamolodchikov model

• Type III Factors

• Vertex Models in Statistical Mechanics

• Quantum Groups

• References

This article studies polynomial invariants in knot theory, their structure and implications for topology, and their relationship with mathematical physics.

We study a Lie bialgebra generated by homotopy classes of loops on an oriented surface F. A biquantization of this Lie bialgebra is constructed in terms of skein equivalence classes of links in F × ℝ. A short overview of related notions of Lie-Poisson Algebra is included.

Elementary exposition of the concept of quantum double is presented and analog of Gauss decomposition for quantum groups is derived.

The following sections are included:

• Introduction

• The Yang-Baxter equation
  • Formulation
  • Examples
  • Braid relations
  • Generalizations

• The classical Yang-Baxter equation
  • Classical limit
  • Universal solution
  • Belavin-Drinfeld theory
  • Examples

• The quantized universal enveloping algebra
  • 'Quantization'
  • The algebra U_q𝔤
  • Representation theory
  • Universal R matrix

• R matrix for the vector representation
  • Linear equations for R
  • Vector representation
  • Centralizer algebras

• The fusion procedure
  • Construction of R matrices
  • The fusion procedure
  • Symmetric tensors

• Face Models
  • Vertex vs. face models
  • Formulation
  • A vertex-face correspondence
  • Vector representation
  • Fusion paths and braid representation

• References

The following sections are included:

• INTRODUCTION

• INFINITESIMAL PURE BRAID RELATIONS

• QUANTIZED UNIVERSAL ENVELOPING ALGEBRA AND R-MATRIX

• FUSION PATH AND NORMALIZED SOLUTIONS

• FUSION METHODS

• ALGEBRAS FACTORING THROUGH THE MONODROMY

• MONODROMY OF n-POINT FUNCTIONS

• References

We present a general theory for construction of link polynomial, topological invariant for knots and links, from an exactly solvable model satisfying the Yang-Baxter relation. First, we present a method to make braid group representation from the Yang-Baxter operator, the constituent of the diagonal-to-diagonal transfer matrix, for the model at criticality. Second, we construct the Markov trace whose existence is a consequence of the crossing symmetry or the second inversion relation satisfied by the solvable model. Third, the general theory is applied to various models, by which a list of new link polynomials are constructed. Lastly, some extensions of the theory are presented. We conclude that a new and powerful approach to knot theory based on the theory of exactly solvable models has been established.

By employing techniques familiar to particle physicists, we develop Kauffman's state model for the Jones polynomial, which uses diagrams looking like Feynman diagrams for scattering, into a systematic, diagrammatic approach to new link polynomials. We systematize the ansatz for S-matrix by symmetry considerations and find a natural interpretation for CPT symmetry in the context of knot theory. The invariance under Reidemeister moves of type III, II and I can be imposed diagrammatically step by step, and one obtains successively braid group representations, regular isotopy and ambient isotopy invariants from Kauffman's bracket polynomials. This procedure is explicitly carried out for the N = 3 and 4 cases, N being the number of particle labels (or charges). With appropriate symmetry ansatz and with annihilation and creation included in the S-matrix, we have obtained link polynomials which generalize the definition of the Akutzu-Wadati polynomials from closed braids to any oriented knots or links with explicit invariance under Reidemeister moves.

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.