纽结理论是数学学科代数拓扑的一个分支，按照数学上的术语来说，是研究如何把若干个圆环嵌入到三维实欧氏空间中去的数学分支。
纽结理论在现代数学中发挥了很大的作用，人们已经在过去的20年中得到了有关这个理论的最有意义的结果。
本书的目的是描述现代纽结理论的主要概念，以及对初学者和专业学者来说都很有用的完整的证明。本书的大部分内容来自作者对虚纽结理论的研究结果。
瓦西里·曼图洛夫，俄罗斯数学家，鲍曼莫斯科国立技术大学几何与拓扑学教授。
Preface
Preface to the second edition
Ⅰ Knots, links, and invariant polynomials
1 Introduction
1.1 Basic definitions
Reidemeister moves. Knot arithmetics
2.1 Polygonal links and Reidemeister moves
2.2 Independence of Reidemeister moves
2.3 Knot arithmetics and Seifert surfaces
3 Links in 2-surfaces in R3. Simplest link invariants
3.1 Knots in 2-surfaces. The classification of torus knots
3.2 The linking coefficient
3.3 The Arf invariant
3.4 The colouring invariant
4 Fundamental group. The knot group
4.1 Digression. Examples of unknotting
4.2 Pundamental group. Basic definitions and examples
4.3 Calculating knot groups
5 The knot quandle and the Conway algebra
5.1 Introduction
5.2 Geometric and algebraic definitions of the knot quandle .
5.2.1 Geometric description of the quandle
5.2.2 Algebraic description of the quandle
5.3 Completeness of the quandle
5.4 Special realisations of the quandle: eolouring invariant, fundamental group, Alexander polynomial
5.5 The Conway algebra and polynomial invariants
5.6 Realisations of the Conway algebra. The Conway-Alexander,Jones, HOMFLY-PT and Kauffman polynomials
5.7 More on Alexander's polynomial. Matrix representation
6 Kauffman's approach to Jones polynomial
6.1 State models in physics and Kauffman's bracket
6.2 Kauffman's forIn of Jones polynomial and skein relations
6.3 Kauffman's two-variable polynomial
7 Properties of Jones polynomials. Khovanov's complex
7.1 Simplest properties
7.2 Tait's first conjecture and Kauffman-Murasugi's theorem
7.3 Menasco-Thistletwaite theorem and the classification of alterhating links
7.4 The third Tait conjecture
7.5 A knot table
7.6 Khovanov's categorification of the Jones polynomial
7.6.1 The two phenomenological conjectures
7.6.2 Spanning tree for Khovanov complex
7.6.3 The Khovanov polynomial and Frobenius extensions
7.6.4 Minimal diagrams of links
8 Lee-Rasmussen invariant, slice knots, and the genus conjecture
8.1 Khovanov homology and Lee homology
8.1.1 Lee's homology
8.1.2 Calculation of Kh'
8.2 The Rasmussen invariant: Definition and basic properties of the invariant
8.2.1 The invariant s
8.2.2 Properties of s
8.3 Behaviour under cobordisms
8.3.1 Elementary cobordisms
8.3.2 Induced maps
8.3.3 Canonical generators
8.3.4 The slice genus
8.4 Computations and relations with other invariants
8.4.1 Using Kh
8.4.2 Positive knots
8.5 R eideIneister moves
Ⅱ Theory of braids
9 Braids, links and representations of braid groups
9.1 Four definitions of the braid group
9.1.1 Geometrical definition
9.1.2 Topological definition
9.1.3 Algebro-geometrical definition
9.1.4 Algebraic definition
9.1.5 Equivalence of the four definitions
……
Ⅲ Vassiliev's invariants.Atoms and d-diagrams
Ⅳ Virtual knots
V Knots, 3-manifolds, and Legendrian knots
D Unsolved problems in knot theory
Bibliography
Index
编辑手记
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