Basis of degree-one Vassiliev cohomology up to order four

Prove that the cohomology classes represented by β₁, β₂, and α₃¹ form a basis over ℚ of degree-one cohomology up to order 4 in the sense of Vassiliev, and that the mod-2 reductions of α₃¹, β₁, and β₂ together with β₃ form a basis over ℤ/2ℤ.

Background

The paper constructs β₁ and β₂ as integer-valued combinatorial 1-cocycles of combinatorial order 4 and β₃ as a nontrivial mod-2 cocycle of combinatorial order 4. Together with Mortier’s cocycle α₃¹, the paper establishes linear independence in the relevant settings and relates the number of these classes to computations on the E₁ page of the Vassiliev spectral sequence.

The authors conjecture that these independent classes exhaust degree-one cohomology through order 4: three classes over ℚ and four classes over ℤ/2ℤ. The conjecture would identify the complete basis in this low-order range, rather than merely provide independent examples.

References

We conjecture that the 1-cohomology classes represented by $\beta_1$ and $\beta_2$, together with $\alpha_31$ form a basis over $\mathbb{Q}$ of the degree-one cohomology up to order 4 in the sense of Vassiliev and ${\alpha_31, \beta_1, \beta_2, \beta_3} \mod2$ form a basis over $\mathbb{Z}/2\mathbb{Z}$.

— Pairings of combinatorial 1-cocycles with loops in knot spaces  (2609.29946 - Zhang, 24 Sep 2026) in Section 1, Introduction, Conjecture immediately following the discussion of the Vassiliev spectral sequence