Non-Regular Lagrangian Concordance

• Non-regular Lagrangian concordance is an exact Lagrangian cobordism between Legendrian submanifolds that cannot be constructed via standard decomposable methods due to the absence of a suitable Weinstein homotopy.
• Its construction employs inversion techniques, the h-principle, and satellite operations to generate examples that highlight the distinction between flexible and rigid cobordisms.
• This phenomenon challenges classical invariants and partial order structures, motivating the development of refined tools in symplectic and contact topology.

Non-regular Lagrangian concordance arises as a central concept in modern symplectic and contact topology, describing situations where exact Lagrangian cobordisms exist between Legendrian submanifolds but fail to possess the rigidity and decomposability of “regular” or “standard” constructions. The paper of non-regular concordance reveals fundamental distinctions between flexible and rigid Lagrangian phenomena, influences the structure of concordance relations, and generates new families of examples that challenge classical classification schemes among Legendrian knots, links, and their higher-dimensional analogues.

1. Definition and Characterization

A Lagrangian concordance is an exact Lagrangian submanifold $L \subset \mathbb{R}_t \times Y^3$ (where $Y^3$ is a contact manifold) that is cylindrical outside a compact region and interpolates between Legendrian submanifolds $\Lambda_-$ and $\Lambda_+$ at $t \to -\infty$ and $t \to +\infty$, respectively. The concordance is called regular if the ambient Weinstein structure can be homotoped so that the Liouville vector field is tangent to $L$ and the Morse function is compatible with the cobordism; equivalently, $L$ can be constructed by standard decomposable moves, i.e., isotopies, standard births/deaths, and surgeries (Rizell et al., 16 Sep 2025).

A non-regular Lagrangian concordance is one for which no such Weinstein homotopy exists—meaning the Lagrangian cannot be made tangent to any Liouville vector field after deformation. Non-regularity thus strictly implies non-decomposability: these concordances cannot be constructed from the elementary building blocks of classical Lagrangian cobordism theory.

In both cases, the classical invariants (such as Thurston–Bennequin number $tb$ and rotation number $r$) are related via

$$tb(\Lambda_+) - tb(\Lambda_-) = -\chi(L), \quad r(\Lambda_-) = r(\Lambda_+)$$

for (totally real and Lagrangian) concordances, with $\chi(L)$ the Euler characteristic (Rizell et al., 30 Aug 2024).

2. Construction Methodology

The construction of non-regular concordances utilizes a combination of topological manipulation, stabilization, and satellite operations:

1. Inversion of Regular Concordance: Given a regular (i.e., decomposable) Lagrangian concordance $L$ from $\Lambda_-$ to $\Lambda_+$ (assumed to be smoothly non-isotopic and fillable), one reflects the concordance in the symplectization coordinate:

$L' = \{(-t, x, y, z) \mid (t, x, y, z) \in L \}.$

This produces a candidate for a reverse cobordism, albeit not generally Lagrangian, but totally real up to homotopy (Rizell et al., 16 Sep 2025).

1. Approximation via the $h$-principle: Applying the $h$-principle for totally real submanifolds, as well as the result that every totally real concordance can be $C^0$-approximated by a Lagrangian concordance with sufficiently stabilized Legendrian ends (Rizell, 29 Aug 2024), one obtains a Lagrangian concordance from a stabilized version $S(\Lambda_+)$ to $S(\Lambda_-)$.
2. Stabilizations and Satellite Operations: The process involves sufficient positive and negative stabilizations to kill rotation numbers and facilitate the application of the $h$-principle. Stabilized ends are then mapped, via a Legendrian satellite construction (notably the Legendrian Whitehead double, denoted $W_0$), to produce augmentable, fillable, non-stabilized Legendrian knots $\Lambda'_+$, $\Lambda'_-$ (Rizell et al., 16 Sep 2025).
3. Transferring Concordances: The satellite operation preserves the Lagrangian concordance relations (Rizell et al., 16 Sep 2025), so the induced satellite concordance $\Sigma(T(L), W_0)$ from $\Lambda'_+$ to $\Lambda'_-$ inherits non-regularity due to the underlying topological obstruction.

3. Comparison: Regular vs Non-regular Concordance

A decomposition of concordances establishes a sharp dichotomy:

Feature	Regular (Decomposable)	Non-Regular (Non-decomposable)
Construction	Handle attachment, standard	Not expressible via elementary moves
Weinstein structure	Can be homotoped to tangency	No such homotopy exists
Inversion property	Invertible via Agol’s result	Inverse is non-regular
Partial order	Satisfies antisymmetry	May violate antisymmetry

Regular (decomposable) concordances obey a partial order structure, as proved using gauge-theoretic and ribbon concordance results (Agol’s theorem). In contrast, the existence of non-regular concordances leads to a failure of antisymmetry: non-isotopic Legendrian fillable knots may each admit a Lagrangian concordance to the other, demonstrated constructively (Rizell et al., 16 Sep 2025, Rizell et al., 30 Aug 2024, Golovko, 18 Nov 2024).

4. Notable Examples and Applications

• Explicit Examples: Beginning with a decomposable filling of the Legendrian mirror $\Lambda_{m(9_{46})}$, the construction yields a decomposable concordance from the standard unknot $U$ to $\Lambda_{m(9_{46})}$. After double-stabilization and Legendrian Whitehead doubling, one obtains a pair of fillable, non-stabilized, smoothly non-isotopic Legendrians with regular and non-regular concordances in opposite directions (Rizell et al., 16 Sep 2025).
• Higher Dimensional Extensions: In dimension $4n+1$ ($n>1$), explicit non-isotopic pairs of closed Legendrians (with equal $tb$ and rotation class) in standard contact vector spaces admit mutual Lagrangian concordances by the flexibility provided by Murphy’s $h$-principle (Golovko, 18 Nov 2024). Such mutual concordances directly contradict antisymmetry for connected Legendrian ends.
• Satellite Functoriality: Legendrian satellite operations, particularly using injective patterns like the Whitehead double, yield new (non-stabilized), fillable Legendrians preserving the essential topological difference and the existence of non-regular concordances (Rizell et al., 16 Sep 2025).

5. Theoretical Implications and Non-Antisymmetry

The existence of non-regular Lagrangian concordances fundamentally disrupts the anticipated partial order structure among Legendrian knots and their generalizations. While regular/decomposable concordances strictly enforce antisymmetry (hence a partial order) via strongly homotopy-ribbon concordance constraints, the non-regular constructions provide a family of counter-examples: for Legendrian fillable knots, there may exist regular concordances in one direction and non-regular ones in the reverse, neither admitting a regular inverse (Rizell et al., 16 Sep 2025, Rizell et al., 30 Aug 2024).

In higher dimensions, even the antisymmetry of the Lagrangian concordance relation fails for closed, connected Legendrians; these findings generalize to exact Lagrangian cobordisms with connected Legendrian ends in $(2n+1)$-dimensions (Golovko, 18 Nov 2024).

6. Broader Significance and Future Directions

Non-regular Lagrangian concordances challenge the robustness of algebraic and topological invariants in distinguishing Lagrangian cobordism types among Legendrian knots. They highlight the need for finer invariants capable of distinguishing regular and non-regular concordances, as well as the necessity of new frameworks for understanding partial ordering in the symplectic and contact settings.

Future research directions include:

• Refining invariants to detect non-regularity beyond classical invariants and Legendrian contact homology.
• Systematically classifying which satellite operations preserve or detect non-regularity.
• Understanding the minimal stabilization required to access the flexible regime and determining to what extent non-regularity is a generic phenomenon among Legendrian fillable knots.
• Extending the framework to higher-dimensional Legendrian submanifolds and exploring ramifications for symplectic field theory and wrapped Fukaya categories.

The constructions and theoretical advances in non-regular Lagrangian concordance fundamentally transform the understanding of the symplectic and contact topology of knots and links, revealing a nuanced interplay between flexibility, rigidity, and algebraic structure.