Representability within Mortier’s combinatorial cohomology framework

Determine whether the 1-cohomology classes represented by the combinatorial 1-cocycles β₁, β₂, and β₃ can be represented by 1-cocycles arising from Mortier’s construction in Reference [Mortier14].

Background

The paper constructs three new combinatorial 1-cocycles, β₁, β₂, and β₃, on the space of long knots using Gauss diagrams with a triangle. The authors compare their construction with Mortier’s theory of combinatorial cohomology, noting that their cocyclicity proof uses Gauss-diagram identities valid for real knots rather than virtual knots, whereas Mortier’s framework has a virtual nature.

The unresolved issue is whether the cohomology classes defined by these new cocycles nevertheless admit representatives within Mortier’s construction. Establishing this would clarify the relationship between the two approaches and determine whether the new classes belong to Mortier’s combinatorial cohomological framework.