An Excursion in Diagrammatic Algebra: Turning a Sphere from Red to Blue (Series on Knots and Everything)

The aim of this book is to give as detailed a description as is possible of one of the most beautiful and complicated examples in low-dimensional topology. This example is a gateway to a new idea of higher dimensional algebra in which diagrams replace algebraic expressions and relationships between diagrams represent algebraic relations. The reader may examine the changes in the illustrations in a leisurely fashion; or with scrutiny, the reader will become familiar and develop a facility for these diagrammatic computations. The text describes the essential topological ideas through metaphors that are experienced in everyday life: shadows, the human form, the intersections between walls, and the creases in a shirt or a pair of trousers. Mathematically informed reader will benefit from the informal introduction of ideas. This volume will also appeal to scientifically literate individuals who appreciate mathematical beauty. Readership: Researchers in mathematics. Contents 18 Preface 10 1. A Sphere 20 2. Surfaces, Folds, and Cusps 24 3. The Inside and Outside 26 4. Dimensions 28 5. Immersed Surfaces 40 6. Movies 48 Double Points and Triple Points 54 Critical Exchanges 57 Example 58 Summary 61 7. Movie Moves 64 The Evolution in the Intrinsic Sphere 64 The Fold Set 65 Double Points and Triple Points 72 Double Points and Folds 80 Triple Points, Double Points, and Folds 88 Conclusion 92 8. Taxonomic Summary 94 9. How Not to Turn the Sphere Inside-out 100 10. A Physical Metaphor 106 Gauss-Morse Codes 110 Summary 111 11. Sarah's Thesis 112 12. The Eversion 116 13. The Double Point and Fold Surfaces 282 Conclusion 288 Bibliography 290 Index 292 J. Scott Carter. Includes Bibliographical References (p. 271) And Index.