Knotted Fields (Lecture Notes in Mathematics, 2344)

This book provides a remarkable collection of contributions written by some of the most accredited world experts in the modern area of Knotted Fields. Scope of the book is to provide an updated view of some of the key aspects of contemporary research, with the purpose to cover basic concepts and techniques commonly used in the context of Knotted Fields. The material is presented to help the interested reader to become familiar with the fundamentals, from fluid flows to electromagnetism, from knot theory to numerical visualization, while presenting the new ideas and results in an accessible way to beginners and young researchers. No advanced knowledge is required, and at the end of each chapter, key references are provided to offer further information on particular topics of interest. All those keen on modern applications of topological techniques to the study of knotted fields in mathematical physics will find here a valuable and unique source of information. The work will be of interest to many researchers in the field. Preface Acknowledgements Contents Contributors 1 A Topological Approach to Vortex Knots and Links 1.1 Helicity and Linking Numbers of Fluid Knots and Links 1.1.1 Linking Number Interpretation of Helicity 1.2 Knot Polynomial Invariants and Helicity 1.2.1 Helicity as an Abelian Chern-Simons Action 1.2.2 Knot Polynomial Invariants in Classical Knot Theory 1.2.2.1 Examples of Polynomial Computation for the Hopf Link and the Trefoil Knot 1.2.3 Derivation of the Adapted Jones Polynomial from Helicity 1.2.4 Derivation of the Adapted HOMFLYPT Polynomial from Helicity 1.2.5 Cascade of Torus Knots and Links in Terms of Adapted Polynomial Values 1.3 Groundstate Energy Spectra of Magnetic Knots and Links 1.4 Optimal Pathway in Knot Polynomial Space 1.4.1 Definition of a Knot Polynomial Space 1.4.2 Unlinking Pathways in Knot Polynomial Space 1.5 Topological Cascade of Borromean Rings 1.5.1 Designing the Initial Condition 1.5.2 Topological Cascade of the Borromean Rings References 2 From Knot Invariants to Knot Dynamics 2.1 Introduction 2.2 Bracket Polynomial and Jones Polynomial 2.2.1 Tensor Networks, Partition Functions and Knot Invariants 2.3 Witten's Work, Quantum Field Theory and Vassiliev Invariants 2.3.1 Vassiliev Invariants and the Jacobi Identity 2.4 Virtual Knot Theory and Extensions of the Jones Polynomial 2.5 Khovanov Homology 2.6 Knot Dynamics 2.6.1 Moving Down Energy Gradients 2.6.2 Hard Knots and Collapsing Tangles 2.6.3 Collapse Examples 2.7 Vortex Knot Reconnection 2.7.1 Spanning Surfaces for Knots 2.7.2 The Seifert Pairing and the Signature 2.7.3 Applications to Vortex Degeneration References 3 Multi-Valued Potentials in Topological Field Theory 3.1 Riemann's Cuts and Multi-Valued Potentials 3.2 Kelvin's Application of Riemann's Cuts 3.2.1 Kelvin's Case Study 3.2.2 Kelvin's Correction to Green's First Identity 3.3 Gauss' Solid Angle Interpretation of the Potential 3.3.1 Interpretation of the Biot-Savart Law in Terms of Solid Angle 3.3.2 Emergence of Topological Linking 3.4 The Impact of Topology on Physics: The Aharonov-Bohm Effect 3.5 Helicity in Multiply Connected Domains 3.6 Kleinert's Multi-Valued Gauge Theory for Singular Fields 3.6.1 Gauge Invariance of Cut Surfaces 3.6.2 Application of Kleinert's Theory to a Circular, Singular Current 3.7 Conclusions References 4 Excitable and Magnetic Knots 4.1 Introduction 4.2 Untangling Excitable Knots 4.3 Stability of Excitable Knots 4.4 Excitable Links 4.5 Colliding Excitable Knots and Links 4.6 Linked Filaments in a Chemical Reaction 4.7 Knots in Frustrated Magnets 4.8 Conclusions References 5 Spiral Waves in Excitable Media: Seifert Framing and Helicity 5.1 Spiral Waves in Excitable Media 5.1.1 Mathematics of Spiral Waves 5.1.2 Spiral Waves in 2D 5.1.3 Spiral Waves in 3D 5.2 Seifert Framing and Individual Helicity of Line Defects 5.2.1 Anti-parallel Reconnection Preserves Seifert Framing, Twist and Writhe References 6 Designing Knotted Fields in Light and Electromagnetism 6.1 What These Lectures Are About 6.2 Complex Scalar Fields, Nodal Filaments, and Design by Random Search 6.2.1 Early Knotted Field Theory 6.2.2 Complex Scalar Fields, Their Nodal Filaments, and Wave Vortices 6.2.3 The Example Fields u and v and Toroidal Coordinates 6.2.4 Tangled Vortex Filaments in Random Waves 6.2.5 Screw Number, Helicity and Threaded Knots 6.3 Knots from Braids, and Design by Algebra 6.3.1 Introduction, Brauner-Milnor Maps for Torus Knots 6.3.2 Generalising the Construction to the Figure-8 Knot 6.3.3 Linked and Knotted Scalar Fields from Simple Braids 6.3.4 Lemniscate Knots and Minimal Braid Words 6.3.5 Creating Knotted Light Beams Experimentally 6.4 Magnetostatics, Solid Angle, and Design by Geometry 6.4.1 Magnetostatics and the Solid Angle Phase Function 6.4.2 Properties of the Solid Angle Phase Function ω 6.4.3 Maxwell's Third Solid Angle Formula, Linking and the Solid Angle Framing 6.5 Designing Null Electromagnetic Fields with Knotted Field Lines 6.5.1 Knotted Textures 6.5.2 The Hopf Fibration and Skyrmionic Hopfions 6.5.3 Helicity and Vector Fields 6.5.4 Towards Relativistic Knotted EM: Null Fields and the Bateman Formalism 6.5.5 The Rañada Hopfion and Knotted Null Fields 6.5.6 Final Remarks References 7 Tangled Vortex Lines: Dynamics, Geometry and Topology of Quantum Turbulence 7.1 Motivation. Is Turbulence Knotted? 7.2 A Brief History of Quantum Fluids 7.3 Quantum Vortices 7.3.1 Quantisation of the Circulation 7.3.2 The Vortex Core 7.3.3 More Insight into the Vortex Core 7.4 Geometry of Quantum Turbulence 7.4.1 In Real Space 7.4.2 In k-Space 7.4.3 Vinen Turbulence 7.5 Dynamics of Quantum Turbulence 7.5.1 Friction 7.5.2 Kelvin Waves 7.5.3 Sound Emission 7.5.4 Reconnections 7.5.5 The Vortex Filament Model 7.6 Topology of Quantum Turbulence 7.6.1 Steady Turbulence in Open Domain 7.6.2 Quantifying the Complexity of the Turbulence 7.6.3 Helicity 7.6.4 Knottiness 7.7 Conclusions References 8 An Introduction to KnotPlot 8.1 Introduction 8.2 Setting It Up 8.3 Loading and Saving 8.4 Changing the View or Embedding 8.5 Relaxing Knots and Links Examples 8.6 Making Pictures 8.7 Making Knot Diagrams 8.8 Creating Knots and Links 8.8.1 Sketching 8.8.2 Editing 8.8.3 Torus Knots and Links 8.8.4 Lissajous Knots 8.8.5 Chains 8.8.6 Braids 8.9 Useful Commands 8.10 Parameter Values 8.11 User Interface 8.12 KnotPlot Distribution 8.12.1 Where the Files Are 8.12.2 Various Catalogues 8.12.3 Knots and Links in Virtual Reality (VR) 8.13 Advanced Dynamics 8.14 Scripting with and Running Without Graphics 8.15 Tabs 8.16 Commands Listed by Activity 8.16.1 Changing the Knot/Link Coordinates 8.16.2 Isolating Components in Links 8.16.3 Combining Multiple Knots/Links 8.16.4 Making a Knot/Link Look Nice 8.16.5 Making Pictures 8.16.6 Relaxing a Knot/Link 8.16.7 Technical Issues and Information 8.16.8 Knot/Link Configuration Measurements 8.16.9 Scripting 8.16.10 Crossing Information 8.16.11 Special Knot Constructions 8.16.12 Safety 8.16.13 File Input/Output References 9 Using the HOMFLYPT Polynomial to Compute Knot Types 9.1 Introduction 9.2 Using Freeware to Compute HOMFLYPT Polynomials and Knot Types 9.2.1 Overview of the Process 9.2.2 Downloading and Installing the Software 9.2.3 Converting HOMFLYPT Polynomials to a Knot Type List Using Versions of jidknot 9.2.4 Computing HOMFLYPT Polynomials Using Jenkins's Algorithm from Crossing Codes Using jhomfly 9.2.5 Simplifying the Crossing Codes Using xinger 9.2.6 Converting 3D Knot Coordinates to Crossing Codes Using coords2egc 9.2.7 Extended Gauss Codes (EGCs) 9.3 Examples 9.3.1 Convert a Knot/Link File into an EGC Using coords2egc 9.3.2 Simplify a File of EGC Codes Using xinger 9.3.3 Compute HOMFLYPT Polynomials from a File of EGCs Using jhomfly 9.3.4 Translate a File of HOMFLYPT Polynomials to a File of Knot Types 9.3.5 Using Piping to String Together the Programs 9.3.6 Helper Scripts 9.3.6.1 Technical Notes 9.3.6.2 Convert a Single Coordinate File to a Knot Type 9.3.6.3 Convert a Number of Coordinate Files to EGCs 9.3.6.4 Convert a Number of Coordinate Files to Knot Types 9.3.6.5 Convert a File of EGCs to Knot Types 9.4 Information About Knot Types Through 16 Crossings 9.4.1 Achiral Prime Knot Types 9.4.2 Chiral Knot Types Whose Mirror Pair Share the Same HOMFLYPT Polynomial 9.4.3 Number of HOMFLYPT Polynomials and Collisions 9.4.4 Peculiarities in the Knot Tables 9.4.5 Counts for Chiral Knot Types 9.5 Resolving Some Types of HOMFLYPT Collisions 9.6 Our libhomfly to Knot Types Conversion File References Index