An integral expression of the first non-trivial one-cocycle of the space of long knots in R3
Abstract
Our main object of study is a certain degree-one cohomology class of the space K of long knots in R3. 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's 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.
0