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A Survey of Knot Theory(1996)
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Akio Kawauchi
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Birkhauser
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first edition
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420
ISBN
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3764351241
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Knot theory is a rapidly developing field of research with many applications not only for mathematics. the present volume, written by a well-known specialist, gives a complete survey of knot theory from its very beginnings to today's most recent research results. The topics include Alexander polynomials, Jones type polynomials, and Vassiliev invariants.
With its appendix containing many useful tables and an extended list of reference with over 3500 entries it is an indispensible book for everyone concerned with knot theory.
The book can serve as an introduction to the field for advanced undergraduate and graduate students. Also researchers working in outside areas such as theoretical physics or molecular biology will benefit from this thorough study which is complemented by many exercises and examples.
Preface
A prelude to the study of knot theory
Notes on research conventions and notations
Chapter 0 Fundamentals of knot theory
0.1 Spaces
0.2 Manifolds and submanifolds
0.3 Knots and links
Supplementary notes for Chapter 0
Chapter 1 Presentations
1.1 Regular presentations
1.2 Braid presentations
1.3 Bridge presentations
Supplementary notes for Chapter 1
Chapter 2 Standard examples
2.1 Two-bridge links
2.2 Torus links
2.3 Pretzel links
Supplementary notes for Chapter 2
Chapter 3 Compositions and decompositions
3.1 Composition of links
3.2 Decompositions of links
3.3 Definition of a tangle and examples
3.4 How to judge the non-splittability of a link
3.5 How to judge the primeness of a link
3.6 How to judge the hyperbolicity of a link
3.7 Non-trivability of a link
3.8 Conway mutation
Supplementary notes for Chapter 3
Chapter 4 Seifert surfaces I: a topological approach
4.1 Definition and existence of Seifert surfaces
4.2 The Murasugi sum
4.3 Sutured manifolds
Supplementary notes for Chapter 4
Chapter 5 Seifert surfaces II: an algebraic approach
5.1 The Seifert matrix
5.2 S-equivalence
5.3 Number-theoretic invariants
5.4 The reduced link module
5.5 The homology of a branched cyclic covering manifold
Supplementary notes for Chapter 5
Chapter 6 The fundamental group
6.1 Link groups and link group systems
6.2 Presentations of a link group
6.3 Subgroups and quotient groups of a link group
Supplementary notes for Chapter 6
Chapter 7 Multi-varibale Alexander polynomials
7.1 The Alexander module
7.2 Invariants of /-module
7.3 Graded Alexander polynomials
7.4 Torres conditions
Supplementary notes for Chapter 7
Chapter 8 Jones type polynomials I: a topological approach
8.1 The Jones polynomial
8.2 The skein polynomial
8.3 The Q and Kauffman polynomials
8.4 Properties of the polynomial invariants
8.5 The skein polynomial via a state model
Supplementary notes for Chapter 8
Chapter 9 Jones type polynomials II: an algebraic approach
9.1 Preliminaries from representation theory
9.2 Link invariants of trace type
9.3 The skein polynomial as a link invariant of trace type
9.4 The Temperley-Lieb algebra
Supplementary notes for Chapter 9
Chapter 10 Symmetries
10.1 Periodic knots
10.2 Freely periodic knots
10.3 Invertible knots
10.4 Amphicherial knots
10.5 Symmetries of a hyperbolic knot
10.6 The symmetry group
10.7 Canonical decompositions and symmetry
Supplementary notes for Chapter 10
Chapter 11 Local transformations
11.1 Unknotting operations
11.2 properties of X -Gordian distance
11.3 Properties of triangle-Gordian distance
11.4 Properties of #-Gordian distance
11.5 Estimation of the X -unknotting number
11.6 Local transformations of links
Supplementary notes for Chapter 11
Chapter 12 Cobordisms
12.1 The knot cobordism group
12.2 The matrix cobordism group
12.3 Link cobordism
Supplementary notes for Chapter 12
Chapter 13 Two-knots I: a topologiocal approach
13.1 A normal form
13.2 Constructing 2-knots
13.3 Seifert hypersurfaces
13.4 Exteriors of 2-knots
13.5 Cyclic covering spaces
13.6 The k -invariant
13.7 Ribbon presentations
Supplementary notes for Chapter 13
Chapter 14 Two-knots II: an algebraic approach
14.1 High-dimensional knot groups
14.2 Ribbon 2-knot groups
14.3 Torsion elements and the deficiency of 2-knot groups
Supplementary notes for Chapter 14
Chapter 15 Knot theory of spatial graphs
15.1 Topology of molecules
15.2 Uses of the notion of equivalence
15.3 Uses of the notion of neighborhood-equivalence
Supplementary notes for Chapter 15
Chapter 16 Vassiliev-Gusarov invariants
16.1 Vassiliev-Gusarov algebra
16.2 Vassiliev-Gusarov invariants and Jones type polynomials
16.3 Knotsevich's iterated integral invariant
16.4 Numerical invariants not of Vassiliev-Gusarov type
Supplementary notes for Chapter 16
Appendix A The equivalence of several notions of link equivalence
Appendix B Covering spaces
B.1 The fundamental group
B.2 Definitions and properties of covering spaces
B.3 The classification of covering spaces
B.4 Covering transformations and the monodromy map
B.5 Branched covering spaces
Appendix C Canonical decompositions of 3-manifolds
C.1 The connected sum
C.2 Dehn's lemma and the loop and sphere theorems
C.3 The equivariant loop and sphere theorems
C.4 Haken manifolds
C.5 Seifert manifolds
C.6 The annulus and torus theorems and the torus decomposition theorem
C.7 Hyperbolic 3-manifolds
Appendix D Heegaard splittings and Dehn surgery descriptions
D.1 Heegaard splittings
D.2 Dehn surgery descriptions
Appendix E The Blanchfield duality theorem
Appendix F Tables of data
F.0 Comments on data
F.1 Knot diagrams
F.2 Type and symmetry
F.3 Knot invariants
F.4 Presentation matrices
F.5 Ribbon presentations
F.6 Skein and Kauffman polynomials
F.7 Surface-link diagrams
References
Index
Introduction to Partial Di...
-Zachmanoglou-
(Dover)
Real Analysis Modern Techn...
-Folland-
(Wiley)
A First Course in Abstract...
-John B. Fraleigh-
(Addison-Wesley)
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QNA
Elementary Differe...
Differential Geome...
Elementary Differe...
Elementary Differe...
Geometric Numerica...
Projective Geometr...
Geometry: from Iso...
Differential Geome...