Cluster structures for smaller cables of twist knots

Determine how to associate cluster structures to the augmentation varieties of smaller cables of twist knots that are Lagrangian fillable but are not isotopic to braid closures satisfying the braid-variety hypotheses used in the paper.

Background

The paper constructs cluster structures for two (2,6)(2,6)-cables of the twist knot m(72)m(7_2) by expressing them as closures of positive braids satisfying the required half-twist and Demazure-product conditions.

The authors point out that smaller cables are also Lagrangian fillable, but do not fit the existing braid-variety framework. Thus, the method does not presently explain how to construct cluster structures for those augmentation varieties.

References

While smaller cables of twist knots (e.g. $n$-copies) are Lagrangian fillable, they are not isotopic to a braid of the form considered in loc. cit. and it is not immediately clear how to associate a cluster structure to their augmentation varieties.

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Section 1, subsection “Comparing Legendrian representatives with infinitely many fillings”