Identify twisting directions for the eight Borromean-rings involution models

Determine which choices of twisting directions on the three components of the Borromean rings produce each of the eight algebraically distinct models of the link involution, potentially by working over the integers using the methods of Alish نظر–Manolescu–Zemke.

Background

The paper constructs eight algebraically distinct models for the link involution of the Borromean rings. These models arise from the possible choices of direction for performing a half twist on each of the three link components, but the computation does not identify which geometric twisting choice corresponds to which algebraic model.

The authors conjecture that this correspondence could be determined by carrying out the construction over the integers, following the ideas cited as AM:HFZ. Establishing this correspondence would remove the remaining ambiguity in the description of the Borromean-rings link involution.

References

We do not sort out which direction of twist produces which model of link involution. We conjecture that one could determine which twisting directions give which model of the link involution by working over $Z$ using the ideas of .

— Link Floer homology, nonformality, and the Borromean rings  (2608.25295 - Hendricks et al., 26 Aug 2026) in Section 1, Introduction, paragraph discussing the link involution

We conjecture that these sets of maps correspond to the orbits of the set of eight orientations on the Borromean rings produced by rotation.

— Link Floer homology, nonformality, and the Borromean rings  (2608.25295 - Hendricks et al., 26 Aug 2026) in Section 6, Remark following the proof of Theorem \ref{thm:iota}