An integral expression of the first nontrivial one-cocycle of the space of long knots in R(3)

Our main object of study is a certain degree-one cohomology class of the space K(3) of long knots in R(3). We describe this class in terms of graphs and configuration space integrals, showing the vanishing of some anomalous obstructions. To show that this class is not zero, we integrate it over a cycle studied by Gramain. As a corollary, we establish a relation between this class and (R-valued) Casson\u27s knot invariant. These are R-versions of the results which were previously proved by Teiblyum, Turchin and Vassiliev over Z/2 in a different way from ours