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Surfaces in 4-Space, written by leading specialists in the field, discusses knotted surfaces in 4-dimensional space and surveys many of the known results in the area. Results on knotted surface diagrams, constructions of knotted surfaces, classically defined invariants, and new invariants defined via quandle homology theory are presented. The last chapter comprises many recent results, and techniques for computation are presented. New tables of quandles with a few elements and the homology groups thereof are included.
This book contains many new illustrations of knotted surface diagrams. The reader of the book will become intimately aware of the subtleties in going from the classical case of knotted circles in 3-space to this higher dimensional case.
As a survey, the book is a guide book to the extensive literature on knotted surfaces and will become a useful reference for graduate students and researchers in mathematics and physics.
List of contents
1 Diagrams of Knotted Surfaces.- 2 Constructions of Knotted Surfaces.- 3 Topological Invariants.- 4 Quandle Cocycle Invariants.- Epilogue.- Append.- References.
Summary
Surfaces in 4-Space, written by leading specialists in the field, discusses knotted surfaces in 4-dimensional space and surveys many of the known results in the area. Results on knotted surface diagrams, constructions of knotted surfaces, classically defined invariants, and new invariants defined via quandle homology theory are presented. The last chapter comprises many recent results, and techniques for computation are presented. New tables of quandles with a few elements and the homology groups thereof are included.
This book contains many new illustrations of knotted surface diagrams. The reader of the book will become intimately aware of the subtleties in going from the classical case of knotted circles in 3-space to this higher dimensional case.
As a survey, the book is a guide book to the extensive literature on knotted surfaces and will become a useful reference for graduate students and researchers in mathematics and physics.
Additional text
From the reviews:
"The book … is devoted to the theory of knotted surfaces in R4 and possesses all the important features of a book which promises to become a classic. … The authors of the book are among the main founders of this theory and have contributed a great deal to its development. … the book may serve as a good introduction for a more or less experienced reader into the beautiful world of knotted surfaces."
Sergej V. Matveev, Mathematical Reviews, 2005e
"The book treats the theory of knotting of surfaces in 4-space presenting up to date results and research … . Each notion is precisely defined with a short historical account included. The results are gradually introduced, illustrated by examples, and original references are always cited. The reader is advised if a result has a higher dimensional counterpart. The book contains an exhaustive list of references and the index. It represents a nice, useful and reliable encyclopaedic presentation of the above mentioned subject … ."
Ivan Ivanšic, Zentralblatt MATH, Vol. 1078, 2006
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From the reviews:
"The book ... is devoted to the theory of knotted surfaces in R4 and possesses all the important features of a book which promises to become a classic. ... The authors of the book are among the main founders of this theory and have contributed a great deal to its development. ... the book may serve as a good introduction for a more or less experienced reader into the beautiful world of knotted surfaces."
Sergej V. Matveev, Mathematical Reviews, 2005e
"The book treats the theory of knotting of surfaces in 4-space presenting up to date results and research ... . Each notion is precisely defined with a short historical account included. The results are gradually introduced, illustrated by examples, and original references are always cited. The reader is advised if a result has a higher dimensional counterpart. The book contains an exhaustive list of references and the index. It represents a nice, useful and reliable encyclopaedic presentation of the above mentioned subject ... ."
Ivan Ivansic, Zentralblatt MATH, Vol. 1078, 2006
