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Record Nr. |
UNINA9910463460303321 |
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Autore |
Livingston Charles |
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Titolo |
Knot theory [[electronic resource] /] / Charles Livingston |
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Pubbl/distr/stampa |
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Washington, D.C., : Mathematical Association of America, c1993 |
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ISBN |
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Descrizione fisica |
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1 online resource (258 p.) |
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Collana |
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The Carus mathematical monographs ; ; no. 24 |
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Disciplina |
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Soggetti |
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Knot theory |
Electronic books. |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Note generali |
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Description based upon print version of record. |
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Nota di bibliografia |
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Includes bibliographical references (p. 233-237) and index. |
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Nota di contenuto |
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""Front Cover""; ""Knot Theory""; ""Copyright Page""; ""Contents""; ""Acknowledgements""; ""Preface""; ""Chapter 1. A Century of Knot Theory""; ""Chapter 2. What Is a Knot?""; ""Section 1: Wild Knots and Unknottings""; ""Section 2: The Definition of a Knot""; ""Section 3: Equivalence of Knots, Deformations""; ""Section 4: Diagrams and Projections""; ""Section 5: Orientations""; ""Chapter 3. Combinatorial Techniques""; ""Section 1: Reidemeister Moves""; ""Section 2: Colorings""; ""Section 3: A Generalization of Colorability, mod p Labelings"" |
""Section 4: Matrices, Labelings, and Determinants""""Section 5: The Alexander Polynomial""; ""Chapter 4. Geometric Techniques""; ""Section 1: Surfaces and Homeomorphisms""; ""Section 2: The Classification of Surfaces""; ""Section 3: Seifert Surfaces and the Genus of a Knot""; ""Section 4: Surgery on Surfaces""; ""Section 5: Connected Sums of Knots and Prime Decompositions""; ""Chapter 5. Algebraic Techniques""; ""Section 1: Symmetric Groups""; ""Section 2: Knots and Groups""; ""Section 3: Conjugation and the Labeling Theorem""; ""Section 4: Equations in Groups and the Group of a Knot"" |
""Section 5: The Fundamental Group""""Chapter 6. Geometry, Algebra, and the Alexander Polynomial""; ""Section 1: The Seifert Matrix""; ""Section 2: Seifert Matrices and the Alexander Polynomial""; ""Section 3: The Signature of a Knot, and other S-Equivalence Invariants""; ""Section 4: Knot Groups and the Alexander Polynomial""; ""Chapter 7. Numerical Invariants""; ""Section 1: Summary of Numerical Invariants""; |
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""Section 2: New Invariants""; ""Section 3: Braids and Bridges""; ""Section 4: Relations Between the Numerical Invariants""; ""Section 5: Independence of Numerical Invariants"" |
""Chapter 8. Symmetries of Knots""""Section 1: Amphicheiral and Reversible Knots""; ""Section 2: Periodic Knots""; ""Section 3: The Murasugi Conditions""; ""Section 4: Periodic Seifert Surfaces and Edmonds' Theorem""; ""Section 5: Applications of the Murasugi and Edmonds Conditions""; ""Chapter 9. High-Dimensional Knot Theory""; ""Section 1: Defining High-dimensional Knots""; ""Section 2: Three Dimensions from a 2-dimensional Perspective""; ""Section 3: Three-dimensional Cross-sections of a 4-dimensional Knot""; ""Section 4: Slice Knots""; ""Section 5: The Knot Concordance Group"" |
""Chapter 10. New Combinatorial Techniques""""Section 1: The Conway Polynomial of a Knot""; ""Section 2: New Polynomial Invariants""; ""Section 3: Kauffman's Bracket Polynomial""; ""Appendix 1. Knot Table""; ""Appendix 2. Alexander Polynomials""; ""References""; ""Index"" |
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Sommario/riassunto |
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Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots. Livingston guides you through a general survey of the topic showing how to use the techniques of linear algebra to address some sophisticated problems, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject—the Conway, Jones, and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology. |
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