Sutured Transverse Surface Theorem

Determine whether the Transverse Surface Theorem holds, in some form, for a sutured manifold equipped with a tight semiflow.

Background

For closed manifolds, the Transverse Surface Theorem characterizes surfaces that can be isotoped to be almost transverse to a pseudo-Anosov flow. The paper explains that a sutured analogue is needed to obtain a flow version of Gabai’s sutured decomposition sequence. Existing work establishes only a homologous version in the first cut case, so the broader theorem remains unresolved.

References

does Transverse Surface Theorem \autocite{LandryMinskyTaylor2025_TransverseSurfacesAndPseudoAnosovFlows_PUB} still hold, in some form, for a sutured manifold with a tight semiflow?

— Pseudo-Anosov flow and dynamics on guts  (2608.30973 - Huang, 31 Aug 2026) in Section 6, subsection “Characterization of flows on the guts”

One may ask whether we can compute the sutured Thurston face by dynamical infomations.

— Pseudo-Anosov flow and dynamics on guts  (2608.30973 - Huang, 31 Aug 2026) in Section 6, subsection “Sutured version of cones”

Further, one may ask if a dual relation holds for the tight semiflows without freely-homotopic orbits: are the cone spanned by closed orbits and the cone spanned by (almost) transverse surfaces dual?

— Pseudo-Anosov flow and dynamics on guts  (2608.30973 - Huang, 31 Aug 2026) in Section 6, subsection “Sutured version of cones”