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Product Code:  SURV/95 
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eBook ISBN:  9781470413224 
Product Code:  SURV/95.E 
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Hardcover ISBN:  9780821829691 
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Product Code:  SURV/95.B 
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Hardcover ISBN:  9780821829691 
Product Code:  SURV/95 
List Price:  $128.00 
MAA Member Price:  $115.20 
AMS Member Price:  $102.40 
eBook ISBN:  9781470413224 
Product Code:  SURV/95.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821829691 
eBook ISBN:  9781470413224 
Product Code:  SURV/95.B 
List Price:  $253.00 $190.50 
MAA Member Price:  $227.70 $171.45 
AMS Member Price:  $202.40 $152.40 

Book DetailsMathematical Surveys and MonographsVolume: 95; 2002; 313 ppMSC: Primary 57; Secondary 20
Braid theory and knot theory are related via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus, one can use braid theory to study knot theory and vice versa.
In this book, the author generalizes braid theory to dimension four. He develops the theory of surface braids and applies it to study surface links. In particular, the generalized Alexander and Markov theorems in dimension four are given. This book is the first to contain a complete proof of the generalized Markov theorem.
Surface links are studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links.
Included is a table of knotted surfaces with a computation of Alexander polynomials. Braid techniques are extended to represent link homotopy classes. The book is geared toward a wide audience, from graduate students to specialists. It would make a suitable text for a graduate course and a valuable resource for researchers.
ReadershipGraduate students and research mathematicians interested in manifolds, cell complexes, and group theory and generalizations.

Table of Contents

Chapters

0. Basic notions and notation

1. Braids

2. Braid automorphisms

3. Classical links

4. Braid presentation of links

5. Deformation chain and Markov’s theorem

6. Surface links

7. Surface link diagrams

8. Motion pictures

9. Normal forms of surface links

10. Examples (spinning)

11. Ribbon surface links

12. Presentations of surface link groups

13. Branched coverings

14. Surface braids

15. Products of surface braids

16. Braided surfaces

17. Braid monodromy

18. Chart descriptions

19. Nonsimple surface braids

20. 1Handle surgery on surface braids

21. The normal braid presentation

22. Braiding ribbon surface links

23. Alexander’s theorem in dimension four

24. Split union and connected sum

25. Markov’s theorem in dimension four

26. Proof of Markov’s theorem in dimension four

27. Knot groups

28. Unknotted surface braids and surface links

29. Ribbon surface braids and surface links

30. 3Braid 2knots

31. Unknotting surface braids and surface links

32. Seifert algorithm for surface braids

33. Basic symmetries in chart descriptions

34. Singular surface braids and surface links


Reviews

This book presents this surface braid theory in a systematic and well organized manner, and is the first to overview the theory ... A complete proof of an analogue of Markov's theorem is presented, which is made available in print for the first time in this book ... Throughout the book, the description of the material is concise and precise, and illustrations are effective and helpful.
Mathematical Reviews 
The present book gives the only full treatment of the basic results on surface braids, and is likely to become the standard reference for its topic.
Zentralblatt MATH


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Braid theory and knot theory are related via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus, one can use braid theory to study knot theory and vice versa.
In this book, the author generalizes braid theory to dimension four. He develops the theory of surface braids and applies it to study surface links. In particular, the generalized Alexander and Markov theorems in dimension four are given. This book is the first to contain a complete proof of the generalized Markov theorem.
Surface links are studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links.
Included is a table of knotted surfaces with a computation of Alexander polynomials. Braid techniques are extended to represent link homotopy classes. The book is geared toward a wide audience, from graduate students to specialists. It would make a suitable text for a graduate course and a valuable resource for researchers.
Graduate students and research mathematicians interested in manifolds, cell complexes, and group theory and generalizations.

Chapters

0. Basic notions and notation

1. Braids

2. Braid automorphisms

3. Classical links

4. Braid presentation of links

5. Deformation chain and Markov’s theorem

6. Surface links

7. Surface link diagrams

8. Motion pictures

9. Normal forms of surface links

10. Examples (spinning)

11. Ribbon surface links

12. Presentations of surface link groups

13. Branched coverings

14. Surface braids

15. Products of surface braids

16. Braided surfaces

17. Braid monodromy

18. Chart descriptions

19. Nonsimple surface braids

20. 1Handle surgery on surface braids

21. The normal braid presentation

22. Braiding ribbon surface links

23. Alexander’s theorem in dimension four

24. Split union and connected sum

25. Markov’s theorem in dimension four

26. Proof of Markov’s theorem in dimension four

27. Knot groups

28. Unknotted surface braids and surface links

29. Ribbon surface braids and surface links

30. 3Braid 2knots

31. Unknotting surface braids and surface links

32. Seifert algorithm for surface braids

33. Basic symmetries in chart descriptions

34. Singular surface braids and surface links

This book presents this surface braid theory in a systematic and well organized manner, and is the first to overview the theory ... A complete proof of an analogue of Markov's theorem is presented, which is made available in print for the first time in this book ... Throughout the book, the description of the material is concise and precise, and illustrations are effective and helpful.
Mathematical Reviews 
The present book gives the only full treatment of the basic results on surface braids, and is likely to become the standard reference for its topic.
Zentralblatt MATH