內容大鋼
這是一本生動的關於數學紐結的說明書，將吸引各式各樣的讀者，從尋求傳統學習以外經驗的本科生，到想要悠閑介紹這個主題的數學家。剛開始深入學習的研究生將會發現這是一份有價值的概述，讀者只需要線性代數的訓練，就可以理解書中提到的數學內容。當從線性代數和基本的群論中引入工具研究紐結的性質時，拓撲和代數之間的相互作用，即代數拓撲，在書中早早出現。
Livingston帶領讀者通過這個課題的概覽，展示如何使用線性代數的技巧來解決一些複雜問題，包括數學中最美麗的主題之一：對稱。本書最後討論了高維紐結理論，以及該學科的一些最新進展，包括Conway、Jones和Kauffman的多項式。補充部分介紹了基本群，它是代數拓撲的核心。
目錄
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
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
[