Nontrivial classes in $H^{\ast }(Imb(S^1,{\mathbb{R}}^n))$ from nontrivalent graph cocycles

Abstract

We construct nontrivial cohomology classes of the space $Imb(S^1,{\mathbb{R}}^n)$ of imbeddings of the circle into ${\mathbb{R}}^n$, by means of Feynman diagrams. More precisely, starting from a suitable linear combination of nontrivalent diagrams, we construct, for every even number $n\ge 4$, a de Rham cohomology class on $Imb(S^1,{\mathbb{R}}^n)$. We prove nontriviality of these classes by evaluation on the dual cycles.

10 pages, 11 figures. V2: minor changes, typos corrected

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• Higher-dimensional BF theories in the Batalin-Vilkovisky formalism: The BV action and generalized Wilson loops
• Configuration spaces and Vassiliev classes in any dimension
• Loop observables for BF theories in any dimension and the cohomology of knots
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Cited by in corpus (10)

• Non-trivalent graph cocycle and cohomology of the long knot space
• Deloopings of the spaces of long embeddings
• Configuration space integrals for embedding spaces and the Haefliger invariant
• An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3
• Knotted families from graspers
• Configuration space integrals and the topology of knot and link spaces
• A Geometric Homology Representative in the Space of Long Knots
• A Kontsevich integral of order 1
• A survey of Bott-Taubes integration
• Cocycles of the space of long embeddings and BCR graphs with more than one loop