Geometry & Topology Monographs

Volume 4: Invariants of knots and 3-manifolds (Kyoto 2001)

T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts and V Turaev (editors)


Kyoto Logo The workshop and seminars on "Invariants of Knots and 3-Manifolds" took place at the Research Institute for Mathematical Sciences (RIMS), Kyoto University, in September 2001. The workshop was held over the period September 17-21. Seminars were held on the Tuesdays, Wednesdays and Thursdays of the other weeks of September, including "Goussarov day" on September 25.
Since the interaction between geometry and mathematical physics in the 1980s, many invariants of knots and 3-manifolds have been discovered and studied: polynomial invariants such as the Jones polynomial, Vassiliev invariants, the Kontsevich invariant of knots, quantum and perturbative invariants, the LMO invariant and finite type invariants of 3-manifolds. The discovery and analysis of the enormous number of these invariants yielded a new area: the study of invariants of knots and 3-manifolds (from another viewpoint, the study of the sets of knots and 3-manifolds). There are also developing topics related to other areas such as hyperbolic geometry via the volume conjecture and the theory of operator algebras via invariants arising from 6j-symbols. On the other hand, recent works have almost completed the topological reconstruction of the invariants derived from the Chern-Simons field theory.
An aim of the workshop and seminars was to discuss future directions for this area. To discuss these matters fully, we planned 1 month of activities, relatively longer than usual. Further, to encourage discussions among the participants, we arranged a short problem session after each talk, and requested the speaker to give his/her open problems there. Many interesting problems were presented in these problem sessions and, based on them, we had valuable discussions in and between seminars and the workshop. Open problems discussed there were edited and formed into a problem list, which, I hope, will clarify the present frontier of this area and assist readers when considering future directions.
The workshop and seminars were organized by T. Ohtsuki, H. Murakami, J. Murakami, and T. Kohno. This proceedings volume was edited by T. Ohtsuki, T. Kohno, T. Le, J. Murakami, J. Roberts, and V. Turaev. The workshop and seminars were partially supported by Grand-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology, Japan: Kiban(C)13640064 (T. Ohtsuki), Kiban(B)12440014 (T. Kohno), Kiban(B)12440015 (S. Kojima), Kiban(B)12440018 (H. Murakami), and partially supported by the Inoue Foundation for Science. The workshop and seminars were held as partial activities of the 2001 RIMS project research "Low-dimensional topology in the twenty-first century", organized by H. Murakami, T. Kobayashi, J. Murakami, T. Ohtsuki, K. Saito, M. Sakuma, and K. Taniyama. The logo was designed by N. Okuda.

1. Pages 1-12 : Bracelets and the Goussarov Filtration of the Space of Knots

2. Pages 13-28 : QHI, 3-manifolds scissors congruence classes and the volume conjecture

3. Pages 29-41 : A homological definition of the Jones polynomial

4. Pages 43-54 : Periodicity of Goussarov-Vassiliev knot invariants

5. Pages 55-68 : On the quantum sl_2 invariants of knots and integral homology spheres

6. Pages 69-87 : Quantum invariants of Seifert 3-manifolds and their asymptotic expansions

7. Pages 89-101 : Polynomial invariants and Vassiliev invariants

8. Pages 103-117 : Knot invariants derived from quandles and racks

9. Pages 119-141 : p-Modular TQFT's, Milnor torsion and the Casson-Lescop invariant

10. Pages 143-160 : Loop spaces of configuration spaces and finite type invariants

11. Pages 161-181 : A surgery formula for the 2-loop piece of the LMO invariant of a pair

12. Pages 183-199 : On configuration space integrals for links

13. Pages 201-214 : Matrix-tree theorems and the Alexander-Conway polynomial

14. Pages 215-233 : Cubic complexes and finite type invariants

15. Pages 235-244 : Power sums and Homfly skein theory

16. Pages 245-261 : Asymptotics and 6j-symbols

17. Pages 263-279 : Generalisations of Rozansky-Witten invariants

18. Pages 281-294 : (2+1)-dimensional topological quantum field theory with a Verlinde basis and Turaev-Viro-Ocneanu invariants of 3-manifolds

19. Pages 295-302 : Torsions of 3-manifolds

20. Pages 303-311 : On the potential functions for the hyperbolic structures of a knot complement

21. Pages 313-335 : Skein module deformations of elementary moves on links

22. Pages 337-362 : The algebra of knotted trivalent graphs and Turaev's shadow world

23. Pages 363-376 : Some computational results on mod 2 finite-type invariants of knots and string links

24. Pages 377-572 : Problems on invariants of knots and 3-manifolds

Publication of this monograph is now complete.

GT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to