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Our main object of study is a certain degree-one cohomology class of thespace K of long knots in R^3. We describe this class in terms of graphs andconfiguration space integrals, showing the vanishing of some anomalousobstructions. To show that this class is not zero, we integrate it over a cyclestudied by Gramain. As a corollary, we establish a relation between this classand (R-valued) Casson's knot invariant. These are R-versions of the resultswhich were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in adifferent way from ours.

An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3