Knotted surfaces and their diagrams / J. Scott Carter, Masahico Saito.

Format:

Book

Author/Creator:

Carter, J. Scott, author.

Saito, Masahico, 1959- author.

Series:

Mathematical surveys and monographs ; no. 55.

Mathematical surveys and monographs, 0076-5376 ; volume 55

Language:

English

Subjects (All):

Knot theory.

Surfaces.

Physical Description:

1 online resource (273 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1998]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Preface""; ""Chapter 1. Diagrams of Knotted Surfaces""; ""1.1. Classical knot diagrams""; ""1.2. Knotted surface diagrams""; ""1.3. Reidemeister moves of classical knots""; ""1.4. Movies of knotted surfaces""; ""1.5. Charts of knotted surfaces""; ""1.6. Examples: how to draw charts and decker curves""; ""1.7. Symbolic presentations of classical knots""; ""1.8. Sentences of knotted surfaces""; ""1.9. Other diagrammatic methods""; ""Chapter 2. Moving Knotted Surfaces""; ""2.1. Equivalence of knotted surfaces""; ""2.2. Roseman moves""; ""2.3. Movie moves""; ""2.4. Chart moves""

""2.5. The grammar of knotted surfaces""""2.6. Singularities of knotted surface isotopies""; ""2.7. Coffee break""; ""Chapter 3. Braid Theory in Dimension Four""; ""3.1. Classical braid theory""; ""3.2. Surface braids""; ""3.3. Charts of surface braids""; ""3.4. Braid movies""; ""3.5. Moves for charts and braid movies""; ""3.6. Homotopy interpretations""; ""Chapter 4. Combinatorics of Knotted Surface Diagrams""; ""4.1. Orientations of the double and triple decker set""; ""4.2. Surfaces in 3-space that do not lift""; ""4.3. Smoothing triple points""

""4.4. Normal Euler numbers and branch points""""4.5. Formulas for colored triple points""; ""4.6. Some combinatorics of charts and sentences""; ""Chapter 5. The Fundamental Group and the Seifert Algorithm""; ""5.1. Wirtinger presentations for classical knots""; ""5.2. Wirtinger presentations for knotted surfaces""; ""5.3. The Alexander module""; ""5.4. A Seifert algorithm for knotted surfaces""; ""Chapter 6. Algebraic Structures Related to Knotted Surface Diagrams""; ""6.1. Generalizations of the Yang-Baxter equation""; ""6.2. Category theory of knotted surfaces""; ""6.3. Conclusion""

""Bibliography""""Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""K""; ""L""; ""M""; ""N""; ""O""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Y""; ""Z""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 243-255) and index.

Description based on print version record.

ISBN:

1-4704-1283-7