Smooth-isotopy invariance of cluster structures

Prove that if a Legendrian knot $\Lambda$ has a 2-graded augmentation variety $Aug_2(\Lambda)$ admitting a cluster structure induced by orientable fillings, then every smoothly isotopic Legendrian knot $\Lambda'$ has a quasi-equivalent cluster structure on $Aug_2(\Lambda')$, with the embedded orientable fillings of $\Lambda$ and $\Lambda'$ in bijection with the cluster seeds.

Background

The paper proves quasi-equivalence of cluster structures for several pairs of smoothly equivalent Legendrian representatives, including representatives of double twist knots and cables. These examples provide evidence that the cluster structure may depend only on the underlying smooth knot type, at least when it is induced by orientable fillings.

The conjecture also asserts a bijection between embedded orientable fillings and cluster seeds, extending an earlier conjectural picture from the Maslov-0 setting to fillings with arbitrary Maslov number.

References

Let $\Lambda$ be a Legendrian knot and suppose $Aug_2(\Lambda)$ admits a cluster structure induced by orientable fillings of $\Lambda$. If $\Lambda'$ is smoothly isotopic to $\Lambda$, then $Aug_2(\Lambda')$ admits a cluster structure quasi-equivalent to that of $Aug_2(\Lambda).$ Moreover, the embedded orientable fillings of $\Lambda$ and $\Lambda'$ are both in bijection with the cluster seeds of $Aug_2(\Lambda).$

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Conjecture 1, Section 1, subsection “Smooth isotopy invariance of cluster structures”