Relate longitude evaluations to leading-term invariants

Establish whether evaluating degree- polynomial functions on link longitudes is equivalent to determining the degree- leading-term invariant of the link.

Background

The paper interprets bar cocycles as polynomial functions on the fundamental group of a link complement, generalizing the Magnus-expansion viewpoint underlying Milnor invariants. The authors suggest that evaluating these functions on longitudes may provide an alternative way to detect the information encoded by the leading-term bar invariant.

The proposed equivalence would clarify the relationship between the bar-cohomological construction and longitude-based link invariants, while potentially making computations easier in settings where longitude evaluation is more accessible than direct manipulation of the leading-term invariant.

References

We conjecture that evaluation of degree $n$ polynomial functions on longitudes could be equivalent to knowledge of the degree $n+1$ leading term invariant, but based on this example expect easier application of longitude evaluation in some settings.

— Bar cohomology of links: beyond Milnor invariants  (2609.11009 - Friedman et al., 10 Sep 2026) in Introduction, Section 1, subsection “Alternate perspectives and future work”