Configuration-space-integral realization of the new cocycles

Determine whether the new combinatorial 1-cocycles β₁, β₂, and β₃ can be expressed by configuration-space integrals in the manner of the 1-cocycle I(Γ) constructed by Sakai and further studied by Kanou and Sakai.

Background

The paper constructs β₁, β₂, and β₃ through Gauss-diagram formulae with a triangle, while related 1-cocycles have been constructed using configuration-space integrals. In particular, the authors cite Sakai’s 1-cocycle I(Γ) and related work by Kanou and Sakai.

The unresolved problem is to determine whether the new combinatorial cocycles admit analogous configuration-space-integral descriptions. Such a realization would link the paper’s diagrammatic constructions to the analytic and geometric methods used for configuration-space integrals.