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Transactions of the American Mathematical Society
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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Additional Information

Michael Hutchings
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: hutching@math.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02213-2
PII: S 0002-9947(98)02213-2
Received by editor(s): May 19, 1995
Additional Notes: Supported by a National Science Foundation Graduate Fellowship.
Article copyright: © Copyright 1998 Michael Hutchings