Weight systems for Mortier’s Kontsevich integral of 1-cocycles

Determine whether the combinatorial 1-cocycles constructed in the paper give weight systems for Mortier’s Kontsevich integral for 1-cocycles described in Reference [Mortier22].

Background

Mortier developed a Kontsevich integral specifically for 1-cocycles. The paper’s newly constructed cocycles β₁, β₂, and β₃ are defined combinatorially by Gauss-diagram formulae with triangles and exhibit nontrivial pairings with canonical loops, including bracket and half-bracket loops.

The authors leave unresolved whether these cocycles are detected by, or can be interpreted as, weight systems for Mortier’s Kontsevich-integral construction. A positive answer would connect the explicit Gauss-diagram cocycles to a broader algebraic-integral framework for finite-type 1-cocycles.