Fundamental-groupoid framework for link invariants

Determine whether the groupoid-ring and algebroid constructions for multivariate Alexander quandles extend to a more general framework based on the fundamental groupoid of a link complement, and whether such a framework yields new link invariants that retain both knotting and linking information.

Background

The paper’s groupoids encode independent deck groups associated with link components, but they are disjoint unions of delooping groupoids rather than fundamental groupoids of link complements. The authors therefore question whether the construction has a deeper topological interpretation using Brown’s fundamental groupoid.

Such an extension could provide a basepoint-free formulation of covering-space theory and potentially preserve more information than the Milnor link group, which the paper notes forgets all knotting data. The authors explicitly identify unknown link invariants native to groupoid-based frameworks as an unresolved direction.

References

Then, there is a deeper question of whether the groupoid and $Z$-algebroid constructions here are indicative of a more general framework based on a fundamental groupoid of the link complement. The utility of the fundamental groupoid has long been championed by R. Brown , as a geometric foundation for homotopy theory. In the context of link topology, our constructions here are suggestive of a basepoint-free formulation of covering space theory, which may possibly involve Brown's fundamental groupoid. Additionally, in contrast to Milnor's link group , which forgets all knotting data, a link groupoid, might presumably be a means to encode both, knotting and linking invariants. This raises the interesting question of yet other unknown link invariants that may be native to groupoid-based frameworks, such as the fundamental groupoid.

— Multivariate Quandles as Groupoid Invariants  (2609.30262 - Arsiwalla et al., 24 Sep 2026) in Section 8, Conclusions and Discussion