Transactions of the American Mathematical Society

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ISSN 1088-6850(e) ISSN 0002-9947(p)

Integration of singular braid invariants and graph cohomology

Author(s): 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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