Maslov-$m$ fillability by smooth knot type

Determine, for each fixed even integer $m$, which smooth knot types admit Legendrian representatives with Maslov-$m$ exact Lagrangian fillings.

Background

The paper constructs exact Lagrangian fillings with several nonzero even Maslov numbers and characterizes important examples among double twist knots. It also proves that every Maslov-0 fillable representative in the studied family is isotopic to the (−1)(-1)-closure of a positive braid.

These results motivate a broader classification problem: identifying the smooth knot types for which at least one Legendrian representative admits an exact filling with a prescribed Maslov number. The question is explicitly posed for every fixed even mm.

References

For a fixed $m\in 2Z$, determine the smooth knot types that have Legendrian representatives admitting Maslov-$m$ exact Lagrangian fillings.

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Question 1, Section 1, subsection “Braid representatives of $[w,c]$”