Completeness of the maximal-Thurston–Bennequin representative collection

Establish whether the collection of maximal-Thurston–Bennequin Legendrian representatives of the double twist knots described by words of the form $[(ZS)^{z^+}(SZ)^{z^-}(SS)^{m/2},c]$ is complete.

Background

The paper studies a family of maximal-Thurston–Bennequin Legendrian representatives of double twist knots, encoded by words in the tangles ZZ and SS. The authors distinguish many such representatives using Legendrian invariants, but state that a complete classification of all Legendrian representatives of double twist knots is not yet available.

The displayed family is therefore described only as conjecturally complete, leaving open whether every maximal-Thurston–Bennequin representative in the relevant family is represented by one of these words.

References

While there is not yet a complete classification of the Legendrian representatives of double twist knots, we can apply the methods of to distinguish Legendrian representatives of $[w, c]$, providing a conjecturally complete collection of max-tb representatives of the form $[(ZS){z+}(SZ){z-}(SS){\frac{m}{2}, c]$ for non-negative integers $z+,$ $z-$, and $m=(-b-1)-2(z++z-)$.

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Section 1, Main results

It is not known whether there exist Maslov-0 Lagrangian fillable Legendrian representatives of the same knot type that are not related by swapping $SZ$ with $ZS$ (or vice versa).

— Cluster structures from Legendrian double twist knots: a comparative approach  (2610.02086 - Hughes et al., 1 Oct 2026) in Remark following Proposition 3.4, Section 3, subsection “Augmentation varieties of Legendrian double twist knots”