A Survey of Knot Theory

Front Cover
Birkhäuser, Dec 6, 2012 - Mathematics - 423 pages

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 references with over 3,500 entries it is an indispensable 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.

 

Contents

Presentations
7
Standard examples
21
Compositions and decompositions
31
a topological approach
47
an algebraic approach
61
The fundamental group
73
Multivariable Alexander polynomials
87
a topological approach
99
a topological approach
171
an algebraic approach
189
Knot theory of spatial graphs
201
VassilievGusarov invariants
209
Appendix A The equivalence of several notions of link equivalence
221
Canonical decompositions of 3manifolds
233
Heegaard splittings and Dehn surgery descriptions
241
Appendix F Tables of data
253

an algebraic approach
113
Symmetries
121
Local transformations
141
Cobordisms
155
References
305
Index
415
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