Recovering lozenge counts from contact geometry

Determine whether the number of dividing curves at infinity for an exhaustion of the cyclic cover associated with a periodic orbit recovers the number of lozenges in the corresponding orbit-space configuration.

Background

The paper establishes that, in the relevant periodic setting, the number of dividing curves is at least the number of lozenges. The unresolved question is whether equality or another precise recovery principle holds, which would characterize the skew property purely through the associated bi-contact structures.

References

In the periodic case, does the number of dividing curves (for an exhaustion of $M\gamma$) at infinity recover the number of lozenges?

— On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry  (2609.03700 - Bowden et al., 3 Sep 2026) in Section 5, Outlook: Further Questions, Question “Detecting Lozenges via bi-contact structure”