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

Abstract

Our main object of study is a certain degree-one cohomology class of the space K 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'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.

11 pages, 4 figures

References in corpus (1)

• Algebraic structures on graph cohomology

Cited by in corpus (2)

• Cocycles of the space of long embeddings and BCR graphs with more than one loop
• The Fox-Hatcher cycle and a Vassiliev invariant of order three