Tying Light in Knots

Table of Contents Contents 1 Topology and Physics: A Historical Overview 1.1 Introduction: Searching for Holes in Fields of Light 1.2 Topology and Physics 1.2.1 Dirac monopoles 1.2.2 Aharanov-Bohm effect 1.2.3 Topology in optics 2 Electromagnetism and Optics 2.1 Electromagnetic fields 2.2 Electromagnetic potentials and gauge invariance 2.3 Linear and Nonlinear Optical Materials 2.4 Polarization and the Poincare sphere 3 Characterizing Spaces 3.1 Loops, holes, and winding numbers 3.2 Homotopy classes 4 Fiber Bundles, Curvature, and Holonomy 4.1 Manifolds 4.2 Vectors and Forms 4.3 Curvature 4.3.1 One-dimension: Curves 4.3.2 Two-dimensions and beyond 4.4 Connections and covariant derivatives 4.5 Fiber Bundles 4.6 Connection and curvature in electromagnetism and optics 5 Topological invariants 5.1 Euler characteristic 5.2 Winding number 5.3 Index 5.4 Chern Numbers 5.5 Linking Number and Other Invariants 6 Vortices and Corkscrews: Singular Optics 6.1 Optical Singularities 6.2 Optical angular momentum 6.3 Vortices and Dislocations 6.4 Knotted and Braided Vortex Lines 6.5 Polarization Singularities 6.6 Optical Mobius strips 7 Optical Solitons 7.1 Solitary waves 7.2 Solitons in optics 8 Geometric and Topological Phases 8.1 The Pancharatnam Phase 8.2 Berry Phase in Quantum Mechanics 8.3 Geometric Phase in Optical Fibers 8.4 Holonomy Interpretation 9 Topological States of Matter and Light 9.1 The Quantum Hall Effect 9.2 Topological Phases and Localized Boundary States 9.3 Topological Photonics A Appendices A.1 Point-Set Topology: Basic Definitions And Results A.2 Brief Review of Group Theory