Extend the leading-term invariant to links in arbitrary 3-manifolds

Construct and establish a leading-term invariant for links in arbitrary 3-manifolds, extending the leading-term invariant developed for links in homology 3-spheres.

Background

The paper constructs its leading-term bar invariant for links in homology 3-spheres, where Alexander duality supplies a canonical basis of first cohomology classes associated with the link components. Extending the construction to arbitrary 3-manifolds would require addressing the additional ambient-manifold homology and the resulting changes in the cohomological marking data.

The authors identify this extension as a future direction and state that a conjectural form of the invariant already exists, making the problem a concrete unresolved generalization of the paper’s principal construction.

References

If we knew the Alexander polynomial can be computed using a rational Seifert surface, we could likely adapt , but we did not find a proof of this assertion in the literature (though see ).

— Welschinger invariants and the Conway polynomial  (2609.26584 - Shende et al., 22 Sep 2026) in Remark immediately following Lemma \ref{lem: also for QHS}

Another planned future direction is to study in more detail the setting of links in arbitrary 3-manifolds, potentially connecting to recent work of Kuzbary and Stees . There we already have a conjectural version of the leading-term invariant.

— 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”