Maximal-Thurston–Bennequin unlink derivative conjecture

Prove that a Legendrian knot with Thurston–Bennequin invariant −1 that admits a Seifert surface with a max-Thurston–Bennequin unlink derivative is decomposably Lagrangian slice.

Background

The paper proves that decomposably Lagrangian slice Legendrian knots admit Seifert surfaces with max-Thurston–Bennequin unlink derivatives. It explicitly leaves the converse as a conjecture because the relevant contact handleslide band sums are not immediately compatible with the standard decomposable-cobordism moves.

References

For subtle reasons that will be discussed later in the article (see \cref{remark:lSGPRC} and \cref{sec:bandsum}), the converse direction (a max-tb unlink derivative implies decomposable sliceness) is more difficult and we content ourselves to leave it as a conjecture.

— Derivative links in contact topology  (2609.30182 - Breen et al., 24 Sep 2026) in Conjecture 1.8, subsection “Derivative characterizations of contact sliceness”