**Mathematical Surveys and Monographs**

Volume: 55;
1998;
258 pp;
Hardcover

MSC: Primary 57;

**Print ISBN: 978-0-8218-0593-0
Product Code: SURV/55**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-1283-8
Product Code: SURV/55.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

# Knotted Surfaces and Their Diagrams

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*J. Scott Carter; Masahico Saito*

In this book the authors develop the theory of knotted
surfaces in analogy with the classical case of knotted curves in
3-dimensional space. In the first chapter knotted surface diagrams
are defined and exemplified; these are generic surfaces in 3-space
with crossing information given. The diagrams are further enhanced to
give alternative descriptions. A knotted surface can be described as a
movie, as a kind of labeled planar graph, or as a sequence of words in
which successive words are related by grammatical changes.

In the second chapter, the theory of Reidemeister moves is
developed in the various contexts. The authors show how to unknot
intricate examples using these moves.

The third chapter reviews the braid theory of knotted
surfaces. Examples of the Alexander isotopy are given, and the braid
movie moves are presented. In the fourth chapter, properties of the
projections of knotted surfaces are studied. Oriented surfaces in
4-space are shown to have planar projections without cusps and without
branch points. Signs of triple points are studied. Applications of
triple-point smoothing that include proofs of triple-point
formulas and a proof of Whitney's congruence on normal Euler
classes are presented.

The fifth chapter indicates how to obtain presentations for the
fundamental group and the Alexander modules. Key examples are worked
in detail. The Seifert algorithm for knotted surfaces is presented and
exemplified. The sixth chapter relates knotted surfaces and
diagrammatic techniques to 2-categories. Solutions to the
Zamolodchikov equations that are diagrammatically obtained are
presented.

The book contains over 200 illustrations that illuminate the
text. Examples are worked out in detail, and readers have the
opportunity to learn first-hand a series of remarkable geometric
techniques.

#### Readership

Graduate students, research mathematicians, physicists, and computer graphics experts interested in knots and links.

#### Reviews & Endorsements

The authors must be congratulated on their heroic endeavors to bring known and unknown results into one book. Who should buy this book? Certainly all topologists with geometric leanings should do so, and their students too.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## Knotted Surfaces and Their Diagrams

- Contents vii8 free
- Preface ix10 free
- Chapter 1. Diagrams of Knotted Surfaces 114 free
- 1.1. Classical knot diagrams 114
- 1.2. Knotted surface diagrams 215
- 1.3. Reidemeister moves of classical knots 1225
- 1.4. Movies of knotted surfaces 1427
- 1.5. Charts of knotted surfaces 1831
- 1.6. Examples: how to draw charts and decker curves 2033
- 1.7. Symbolic presentations of classical knots 3346
- 1.8. Sentences of knotted surfaces 3447
- 1.9. Other diagrammatic methods 3851

- Chapter 2. Moving Knotted Surfaces 4154
- Chapter 3. Braid Theory in Dimension Four 97110
- Chapter 4. Combinatorics of Knotted Surface Diagrams 131144
- Chapter 5. The Fundamental Group and the Seifert Algorithm 169182
- Chapter 6. Algebraic Structures Related to Knotted Surface Diagrams 203216
- Bibliography 243256
- Index 257270