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

TL;DR

This paper describes a specific cohomology class of the space of long knots in R^3 using graph and configuration space integrals, linking it to Casson's knot invariant and building on prior Z/2 results.

Contribution

It introduces a new integral expression for a non-trivial cohomology class of the space of long knots, relating it to classical knot invariants.

Findings

Established the non-vanishing of the cohomology class via integration over Gramain's cycle.

Connected the cohomology class to Casson's knot invariant.

Extended previous Z/2 results to R-valued invariants.

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.