Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots

We show that the family of smoothly non-isotopic Legendrian pretzel knots from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants as the standard unknot have front-spuns that are Legendrian isotopic to the front-spun of the unknot. Besides that, we construct the first examples of Lagrangian concordances between Legendrian knots that are not regular, and hence not decomposable. Finally, we show that the relation of Lagrangian concordance between Legendrian knots is not anti-symmetric, and hence does not define a partial order. The latter two results are based upon a new type of flexibility for Lagrangian concordances with stabilised Legendrian ends.

💡 Research Summary

The paper establishes three new phenomena concerning Legendrian knots and Lagrangian concordances. First, the authors revisit the family of Legendrian pretzel knots (\Lambda_m) (the pretzel knots (P(3,-3,-m)) with (m>3)) constructed by Cornwell‑Ng‑Sivek. These knots are smoothly non‑isotopic but share the same classical invariants (Thurston–Bennequin number (-1) and rotation number (0)) and the same Chekanov–Eliashberg DGA as the standard Legendrian unknot (U). By applying the front‑spinning construction (\Sigma_{S^n}) (which embeds a Legendrian submanifold (\Lambda\subset\mathbb{R}^{2k+1}) into a higher‑dimensional contact space as (\Lambda\times S^n)), the authors prove that for every (n\ge1) the spun Legendrian (\Sigma_{S^n}\Lambda_m) is Legendrian isotopic to (\Sigma_{S^n}U). In other words, after a single suspension the pretzel family becomes Legendrian‑trivial.

Two independent proofs are given. The first uses Legendrian ambient surgery: each (\Lambda_m) can be obtained from two tb = −1 unknots by a Legendrian surgery along an arc (\eta_m). The arc admits a stabilization, and after spinning the stabilized arc (\eta_m\times0_{S^n}) becomes formally Legendrian isotopic to a spun stabilized arc (\eta_{0,m}\times0_{S^n}). By Murph​y’s (h)-principle for loose Legendrians and the ambient surgery theorem, the resulting products (\Lambda_m\times0_{S^n}) and (U\times0_{S^n}) are Legendrian isotopic. The second proof exploits the Weinstein handle‑body picture of the exact Lagrangian cobordism from the unlink to (\Lambda_m). The complement of this cobordism is a Weinstein domain whose handles are attached away from the positive end; after spinning, the complement becomes flexible because the attaching Legendrians are stabilized. A resulting symplectomorphism of the symplectisation identifies the standard Lagrangian fillings of (U\times0_{S^n}) and (\Lambda_m\times0_{S^n}); restricting to the contact boundary yields the desired Legendrian isotopy.

The second major contribution is the construction of the first non‑regular Lagrangian concordances between Legendrian knots in the symplectisation of (\mathbb{R}^3). Starting with any Legendrian knot (\Lambda) that admits a decomposable exact Lagrangian filling (for instance the non‑trivial knot (9_{46})), the authors apply a large number (k) of both positive and negative stabilizations to obtain a new Legendrian (\Lambda^-). Using recent work of the first author, a totally real concordance between (\Lambda^-) and the standard unknot (U) can be approximated by an exact Lagrangian concordance (C\subset(\mathbb{R}_t\times\mathbb{R}^3,d(e^t\alpha_0))). The negative end (\Lambda^-) is stabilized, hence it does not admit any Lagrangian filling, while the positive end is the unknot stabilized the same number of times. The authors prove that this concordance is not regular: regularity would require the Weinstein complement to be a Weinstein domain with the Lagrangian as a skeleton, which is impossible because the negative end is loose and the complement would have to carry non‑trivial homology class. The non‑regularity is detected via a gauge‑theoretic obstruction (Kronheimer–Mrowka) and a generalisation of a theorem of Cornwell‑Ng‑Sivek stating that any smooth concordance from a non‑trivial knot to the unknot must have a local maximum, precluding a decomposable (hence regular) Lagrangian concordance. Moreover, the constructed concordance is not a ribbon concordance.

Finally, the paper settles a long‑standing question about the order‑theoretic nature of Lagrangian concordance. The relation (\Lambda_-\preceq_{\text{conc}}\Lambda_+) holds when there exists a Lagrangian concordance from (\Lambda_-) to (\Lambda_+). It is reflexive and transitive, but the authors show it fails to be antisymmetric. Indeed, for a Legendrian (\Lambda) with a decomposable filling, after sufficiently many stabilizations the resulting (\Lambda^-) admits concordances both from and to the unknot (U) (with the same number of stabilizations). However, the converse direction (U\preceq_{\text{conc}}\Lambda^-) is realized by a regular concordance (the filling), while (\Lambda^-\preceq_{\text{conc}}U) is realized only by the non‑regular concordance constructed above. Since a regular concordance cannot exist in this direction (by the gauge‑theoretic obstruction), the relation is not antisymmetric and therefore does not define a partial order on Legendrian isotopy classes.

In summary, the work demonstrates that front‑spinning can erase Legendrian knotting, that Lagrangian concordances with stabilized ends possess a new flexibility leading to non‑regular examples, and that the Lagrangian concordance relation is inherently non‑partial‑ordered. These results bridge Legendrian knot theory, Weinstein handle theory, and gauge theory, opening new avenues for studying high‑dimensional Legendrian submanifolds and their Lagrangian cobordisms.