Contents
Preface ix
Chapter 1. Diagrams of Knotted Surfaces 1
1.1. Classical knot diagrams 1
1.2. Knotted surface diagrams 2
1.3. Reidemeister moves of classical knots 12
1.4. Movies of knotted surfaces 14
1.5. Charts of knotted surfaces 18
1.6. Examples: how to draw charts and decker curves 20
1.7. Symbolic presentations of classical knots 33
1.8. Sentences of knotted surfaces 34
1.9. Other diagrammatic methods 38
Chapter 2. Moving Knotted Surfaces 41
2.1. Equivalence of knotted surfaces 41
2.2. Roseman moves 42
2.3. Movie moves 44
2.4. Chart moves 52
2.5. The grammar of knotted surfaces 75
2.6. Singularities of knotted surface isotopies 78
2.7. Coffee break 88
Chapter 3. Braid Theory in Dimension Four 97
3.1. Classical braid theory 97
3.2. Surface braids 99
3.3. Charts of surface braids 100
3.4. Braid movies 116
3.5. Moves for charts and braid movies 117
3.6. Homotopy interpretations 123
Chapter 4. Combinatorics of Knotted Surface Diagrams 131
4.1. Orientations of the double and triple decker set 131
4.2. Surfaces in 3-space that do not lift 133
4.3. Smoothing triple points 145
4.4. Normal Euler numbers and branch points 148
4.5. Formulas for colored triple points 161
4.6. Some combinatorics of charts and sentences 166
Chapter 5. The Fundamental Group and the Seifert Algorithm 169
5.1. Wirtinger presentations for classical knots 169
5.2. Wirtinger presentations for knotted surfaces 171
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