Integration of singular braid invariants and graph cohomology

Hutchings, Michael

doi:10.1090/S0002-9947-98-02213-2

Integration of singular braid invariants and graph cohomology
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by Michael Hutchings

Trans. Amer. Math. Soc. 350 (1998), 1791-1809

DOI: https://doi.org/10.1090/S0002-9947-98-02213-2

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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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