Knots, links, braids and 3-manifolds: An introduction to the new invariants..(AMS 1997)

This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. It emphasizes the geometric aspects of the theory and treats topics such as braids, homeomorphisms of surfaces, surgery of 3-manifolds (Kirby calculus), and branched coverings. This attractive geometric material, interesting in itself yet not previously gathered in book form, constitutes the basis of the last two chapters, where the Jones-Witten invariants are constructed via the rigorous skein algebra approach (mainly due to the Saint Petersburg school). Unlike several recent monographs, where all of these invariants are introduced by using the sophisticated abstract algebra of quantum groups and representation theory, the mathematical prerequisites are minimal in this book. Numerous figures and problems make it suitable as a course text and for self-study. Contents......Page 4 Foreword......Page 6 §1. The topology of knots and links......Page 10 §2. Tricks with strings and ribbons......Page 19 Comments......Page 25 §3. The Jones polynomial......Page 27 §4. Vassiliev invariants......Page 40 Comments......Page 48 §5. The braid group.......Page 51 §6. The Alexander and Markov theorems......Page 58 §7. Pure braids......Page 65 Comments......Page 69 §8. Heegaard splittings......Page 70 §9. Heegaard splittings for manifolds with boundary......Page 76 §10. Heegaard diagrams......Page 78 §11. Lens spaces......Page 80 Comments......Page 84 §12. The Dehn-Lickorish theorem and its corollaries......Page 85 §13. Proof of the Dehn-Lickorish theorem......Page 92 Comments......Page 95 §14. Rational surgery along trivial knots......Page 97 §15. Linking numbers......Page 102 §16. Integer surgery......Page 105 §17. Lens spaces revisited......Page 110 §18. Homology spheres......Page 111 §19. The Kirby calculus......Page 119 Comments......Page 127 §20. Branched coverings of surfaces......Page 129 §21. Riemann-Hurwitz formula......Page 133 §22. Branched coverings of 3-manifolds......Page 138 §23. Three-manifolds as branched covers of S^3......Page 143 §24. Branched coverings and colored links......Page 154 §25. The Borromeo rings as a universal link......Page 159 Comments......Page 166 §26. The Temperley-Lieb algebra and other skein algebras......Page 167 §27. The Jones-Wentzl idempotent......Page 173 §28. Invariance with respect to the second Kirby move......Page 179 §29. Invariance with respect to the first Kirby move......Page 183 Comments......Page 190 §30. Polynomial invariants of links in RP^3......Page 192 §31. Invariants of framed links in three-manifolds......Page 195 §32. Knots and physics......Page 197 Appendix......Page 205 Solutions......Page 212 References......Page 230 Index......Page 236 An introduction to the work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. It emphasizes the geometric aspects of the theory and discusses topics such as braids, homeomorphisms of surfaces and surgery of 3-manifolds. This text is intended for researchers working in geometry and topology, and covers knots, links, braids and 3-manifolds.