Necessity of the essential, shuttle, or multi-coupled cycle condition for universal red punctures

Determine whether the sufficient condition in Theorem \ref{thm: all-red puncture} is also necessary; specifically, establish whether every string algebra whose every geometric model contains a red puncture must have a quiver containing an essential cycle, a shuttle cycle, or a multi-coupled cycle.

Background

The paper studies saturated labelled tiled-surface models of string algebras and relates red punctures to primitive permitted cycles in saturated locally gentle covers. It proves that an essential cycle, a shuttle cycle, or a multi-coupled cycle in the quiver is sufficient to ensure that every geometric model contains a red puncture.

The unresolved issue is whether these three combinatorial configurations exhaust all possibilities for universal red-puncture behavior. The subsequent Lemma \ref{lem:type-III-elimination} gives partial evidence by proving that, when every saturated locally gentle cover contains a primitive permitted cycle, one can choose such a cycle avoiding all Type (III) non-gentle vertices; however, the lemma does not establish the full necessity of the three configurations.

References

Is the sufficient condition given in Theorem \ref{thm: all-red puncture} also necessary? Although this question has not been fully resolved, the following lemma provides supporting evidence: if every saturated locally gentle cover contains a primitive permitted cycle, then such a cycle can always be chosen to avoid non-gentle vertices of Type~\textup{(III)}.

— On Geometric Models of String Algebras: Uniqueness of Surfaces and Existence of Red Punctures  (2608.14360 - Xin et al., 14 Aug 2026) in Section “Geometric models with red punctures,” immediately following Theorem \ref{thm: all-red puncture}