Tightness of the remaining Li–Mak contact structures

Determine whether the Li–Mak contact structures on the contact plumbed manifolds associated with all graphs in Figure 1 other than $\Gamma_3$, $\Gamma_{7,1}$, and $\Gamma_{7,5}$ are tight.

Background

The paper identifies strong symplectic fillability of the Li–Mak contact structures (YΓ,ξLM)(Y_\Gamma,\xi_{LM}) as a necessary condition for realizing the corresponding pseudofree symplectic prime-order actions. Strong fillability is known for the graphs Γ3\Gamma_3, Γ7,1\Gamma_{7,1}, and Γ7,5\Gamma_{7,5}.

For every other graph in Figure 1, the paper states that even the weaker contact-topological property of tightness has not been determined. Resolving this question would provide obstructions—or potentially support—for realizing the remaining group-action cases.

References

For the rest of the graphs in Figure 1, we do not even know whether $(Y_\Gamma,\xi_{LM})$ is tight or not.

— Symplectic symmetries of a hypothetical exotic $CP^2$: a potential construction or a nonexistence proof  (2610.06746 - Chen, 5 Oct 2026) in Section 1, paragraph following the discussion of Theorem 1.3