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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Integration of singular braid invariants
and graph cohomology

Author: Michael Hutchings
Journal: Trans. Amer. Math. Soc. 350 (1998), 1791-1809
MSC (1991): Primary 57M25; Secondary 20C07
MathSciNet review: 1475686
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Abstract: We prove necessary and sufficient conditions for an arbitrary invariant of braids with $m$ double points to be the ``$m^{th}$ derivative'' of a braid invariant. We show that the ``primary obstruction to integration'' is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on $m$ which works for invariants with values in any abelian group.

We find that to prove our theorem, we must show that every relation among four-term relations satisfies a certain geometric condition. To find the relations among relations we show that $H_1$ of a variant of Kontsevich's graph complex vanishes. We discuss related open questions for invariants of links and other things.

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  • 1. D. Altschuler and L. Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Comm. Math. Phys 187 (1997), 261-287. CMP 97:16
  • 2. V. I. Arnol'd, The Vassiliev theory of discriminants and knots, in Proceedings of the 1992 European Congress of Mathematicians, Vol. 1, Birkhäuser, 1994, pp. 3-29. MR 96m:57010
  • 3. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423-472. MR 97d:57004
  • 4. -, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995), 13-32. MR 96b:57004
  • 5. D. Bar-Natan, S. Garoufalidis, L. Rozansky and D. Thurston, The øArhus invariant of rational homology 3-spheres I: a highly nontrivial flat connection on $S^3$, preprint, q-alg/9706004.
  • 6. D. Bar-Natan and A. Stoimenow, The fundamental theorem of Vassiliev invariants, Geometry and Physics (Aarhus 1995), 101-134, Lecture Notes in Pure and Appl. Math., 184, Dekker, 1997. CMP 97:05
  • 7. J. Birman and X-S. Lin, Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225-270. MR 94d:57010
  • 8. R. Bott and C. H. Taubes, On the self-linking of knots, J. Math. Phys. 35 no. 10 (1994), 5247-5287. MR 95g:57008
  • 9. V. G. Drinfel'd, On quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457. MR 91b:17016
  • 10. S. Garoufalidis and J. Levine, On finite type 3-manifold invariants II, Math. Ann. 306 (1996), 691-718. CMP 97:03
  • 11. S. Garoufalidis and T. Ohtsuki, On finite type 3-manifold invariants III: manifold weight systems, to appear in Topology.
  • 12. A. Hatcher, private communication, 1995.
  • 13. T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Braids (Santa Cruz, CA, 1986), 339-363, Contemp. Math. 78, AMS, 1988. MR 90h:20056
  • 14. M. Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992), 97-121, Progr. Math. 120, Birkhäuser, 1994. MR 96h:57027
  • 15. -, Vassiliev's knot invariants, Adv. Sov. Math. 16 no. 2 (1993), 137-150. MR 94k:57014
  • 16. T. T. Q. Le, J. Murakami and T. Ohtsuki, On a universal invariant of 3-manifolds, preprint, q-alg/9512002.
  • 17. X-S. Lin, Braid algebras, trace modules and Vassiliev invariants, Columbia University preprint, 1994.
  • 18. T. Ohtsuki, Finite type invariants of integral homology 3-spheres, J. Knot Theory Ramifications 5 (1996), 101-115. MR 97i:57019
  • 19. T. Stanford, Finite-type invariants of knots, links, and graphs, Topology 35 (1996), 1027-1050. MR 97i:57009
  • 20. A. Stoimenow, Stirling numbers, Eulerian idempotents and pure braid cohomology, preprint, 1995.
  • 21. D. Thurston, Integral expressions for the Vassiliev knot invariants, Harvard College senior thesis, 1995.
  • 22. V. A. Vassiliev, Complements to Discriminants of Smooth Maps: Topology and Applications, Amer. Math. Soc., 1992. MR 94i:57020
  • 23. P. Vogel, Algebraic structures on modules of diagrams, preprint, 1995.
  • 24. S. Willerton, A combinatorial half-integration from weight system to Vassiliev invariant, to appear in J. Knot Theory Ramifications.

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Additional Information

Michael Hutchings
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138

Received by editor(s): May 19, 1995
Additional Notes: Supported by a National Science Foundation Graduate Fellowship.
Article copyright: © Copyright 1998 Michael Hutchings

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