Equality of combinatorial and finite-type orders

Prove that, for combinatorial 1-cocycles defined by Gauss diagrams with a triangle, the combinatorial order equals the finite-type order in the sense of Vassiliev.

Background

The paper defines the combinatorial order of a 1-cocycle by the maximum number of arrows in its Gauss-diagram configurations and proves finite-type consequences for pairings with canonical loops. It constructs examples of combinatorial order 4 and relates their evaluations to Vassiliev invariants.

The broader unresolved issue is whether the combinatorial filtration defined using Gauss diagrams with a triangle agrees exactly with Vassiliev’s finite-type filtration for all such combinatorial 1-cocycles. This would establish that the diagrammatic notion of order captures the intrinsic finite-type order without discrepancy.

References

For the combinatorial 1-cocycles defined by Gauss diagrams with a triangle, the combinatorial order coincides with the finite-type order in the sense of Vassiliev in .

— Pairings of combinatorial 1-cocycles with loops in knot spaces  (2609.29946 - Zhang, 24 Sep 2026) in Section 1, Introduction, Conjecture labeled Conjecture~\ref{conj:order}