Short layered parents

Establish whether every non-layered veering triangulation with n tetrahedra admits a layered veering parent obtained by drilling out a single flow cycle of length at most n.

Background

The paper proves that every veering triangulation arising from a transitive pseudo-Anosov flow has a layered parent after drilling one additional closed orbit, but the available general bound on the length of a representing flow cycle is very large. Exhaustive computations for triangulations with at most 16 tetrahedra found much shorter cycles, motivating the proposed linear bound.

References

Every non-layered veering triangulation with $n$ tetrahedra admits a layered veering parent obtained by drilling out a single flow cycle of length at most $n$.

— Drilling veering triangulations with applications to pseudo-Anosov flows  (2610.12302 - Parlak et al., 8 Oct 2026) in Section 5.1, Conjecture 5.5 (label conj:layered)

Let $\zeta$ be a flow cycle of a finite veering triangulation $V$. To what extent do combinatorial features of $V$ and $\zeta$, such as those listed above, determine combinatorial features of $V_\zeta$?

— Drilling veering triangulations with applications to pseudo-Anosov flows  (2610.12302 - Parlak et al., 8 Oct 2026) in Section 6.2, Question 6.2 (label question:combinatorics)