Annulus Isotopy for General Decomposition Surfaces

Determine whether an analogue of the Annulus Isotopy Theorem holds for every decomposition surface satisfying suitable conditions.

Background

The second stage in the paper’s construction is the alignment of product annuli with the pseudo-Anosov flow. The author identifies the extension from product annuli to general decomposition surfaces as an unresolved question, because such an extension would provide the second step of a broader flow-version of the sutured decomposition sequence.

References

One may ask if a similar result holds for any decomposition surface with suitable conditions.

— Pseudo-Anosov flow and dynamics on guts  (2608.30973 - Huang, 31 Aug 2026) in Section 1, subsection “Construction”

By our main theorem, both results hold if $A$ is replaced by $A$. However, we do not know whether $A\neq A$.

— The top coefficient of the Links-Gould invariant is Murasugi-sum multiplicative  (2610.09354 - Anghel et al., 7 Oct 2026) in Section 1, Introduction