Legendrian Isotopy Invariants

• Legendrian isotopy invariants are structures that remain unchanged under smooth deformations, crucial for classifying knots, links, and higher-dimensional submanifolds.
• The Chekanov–Eliashberg DGA, microlocal sheaf categories, and combinatorial racks exemplify methodologies that compute these invariants via holomorphic curves, constructible sheaves, and front diagram analyses.
• Recent advances incorporate generating family spectra and persistence barcodes, providing quantitative and computable measures that enrich our understanding of contact topology rigidity and flexibility.

A Legendrian isotopy invariant is a structure, algebraic or categorical, assigned to a Legendrian submanifold in a contact manifold, whose isomorphism class remains unchanged under smooth Legendrian isotopy. Such invariants are central to the classification and paper of Legendrian knots, links, and higher-dimensional submanifolds. They distinguish Legendrian embeddings up to isotopy and capture subtle contact-geometric and topological information not accessible via classical invariants.

1. Differential Graded Algebraic Invariants

The archetypal Legendrian isotopy invariant is the Chekanov–Eliashberg Differential Graded Algebra (DGA). Given a generic Legendrian submanifold $L$ in a cooriented contact manifold $(Y^{2n+1},\xi)$, the DGA $(\mathcal{A}(L),\partial)$ is generated by Reeb chords of $L$ (integral curves of the Reeb vector field starting and ending on $L$), with grading from a Maslov potential and a differential counting rigid holomorphic disks in the symplectization $(\mathbb{R}_s \times Y, d(e^s \alpha))$ with boundary on $\mathbb{R} \times L$ (Ekholm, 2015, Licata et al., 2010, Rizell et al., 2021). The DGA is defined up to stable tame isomorphism—a composition of elementary automorphisms and stabilizations—under Legendrian isotopy, and its homology, Legendrian contact homology (LCH), is thus invariant. This invariance follows via a trace construction using Lagrangian cobordisms associated to the isotopy, producing stable tame isomorphisms at each bifurcation (Rizell et al., 2021).

In Seifert fibered and other non-standard contact manifolds, the DGA must be enriched to accommodate orbifold points and monodromy, with differentials and gradings reflecting the local orbifold geometry. Legendrian isotopy invariance in these settings is established by direct combinatorial analysis of the affected moduli spaces through Reidemeister moves and teardrop isotopies (Licata et al., 2010).

2. Sheaf-Theoretic and Categorical Invariants

A distinct class of invariants arises from microlocal sheaf theory. For a Legendrian $\Lambda^\infty$ in the cosphere bundle $S^*M$ of a manifold $M$, there is an associated dg-derived category $Sh(M,\Lambda^\infty)$ of constructible sheaves with singular support at infinity contained in $\Lambda^\infty$ (Zhou, 2018, Shende et al., 2014). The Guillermou-Kashiwara-Schapira theorem guarantees the quasi-equivalence type of this dg-category is Legendrian isotopy invariant (under convexity and Weinstein thickening conditions), as proved by constructing continuation kernels that quantize the Hamiltonian isotopy (Zhou, 2018, Shende et al., 2014). This sheaf-theoretic category links constructible sheaves, Fukaya categories, and the augmentation category of the Chekanov–Eliashberg DGA, unifying invariants from microlocal, symplectic, and holomorphic curve theories.

3. Combinatorial and Discrete Invariants

Racks and their generalizations provide robust combinatorial isotopy invariants via purely algebraic relations encoding front diagram data. The generalized Legendrian rack (GL-rack) adds structure maps corresponding to up and down cusps and derives Legendrian isotopy invariance via explicit verification of the Legendrian Reidemeister moves' algebraic effects (Karmakar et al., 2023). The Legendrian isotopy class of a link is thus detected by the isomorphism class of its GL-rack presentation, which can distinguish infinitely many Legendrian unknots or trefoils.

In another combinatorial direction, equivalence classes of Morse complex sequences (MCS) and augmentations of the Chekanov–Eliashberg DGA are in bijection for classical Legendrian knots. The cardinality of such MCS classes becomes a Legendrian isotopy invariant, offering a discrete and computable classifier refined beyond classical invariants (Henry et al., 2014).

4. Invariants Derived from Generating Families and Homotopy Types

For a Legendrian admitting a linear-at-infinity generating family, stable homotopy types of certain sublevel spaces become Legendrian isotopy invariants. The construction proceeds via the "difference function" associated to a generating family $f$, with a prespectrum $\{X_i\}$ built from quotients of sublevel sets stabilized under addition of quadratic terms. The spectrification yields a spectrum $C(\Lambda,f)$ whose stable homotopy type is invariant under generating family equivalence and hence under Legendrian isotopy. When extending to Lagrangian fillings, the Seidel isomorphism lifts from generating family homology to this invariant at the spectrum level, introducing new constraints on fillability and fiber dimensions (Tanaka et al., 2 Aug 2024).

5. Isotopy Invariants in Higher Dimensions and Page Crossing Numbers

For Legendrian surfaces in closed contact $5$-manifolds with admissible open books, the maximal sum of Thurston–Bennequin numbers over intersection curves with page doubles, maximized among isotopic representatives, yields the maximal (relative/absolute) page crossing number $M\mathcal{P}_X(L)$ or $M\mathcal{P}_{(B,f)}(L)$. This is a genuine Legendrian isotopy invariant distinguishing surfaces where the classical $tb$ is ineffective, and computable from handle diagrams and Stein page data (Arikan et al., 2021).

6. Invariant Metrics and Persistence Barcodes in Isotopy Classes

Recent developments establish invariant distances on Legendrian isotopy classes—typically constructed on universal covers of isotopy classes and defined in terms of orderability and positive loops. These metrics are invariant under the identity component of the contactomorphism group, discrete except in the presence of additional rigidity, and can be unbounded under appropriate hypotheses (orderability and existence of a contractible positive loop) (Arlove, 24 Jul 2025). The persistence module structure arising from filtered Rabinowitz–Floer complexes also leads to quantitative invariants, where barcodes, persistent over isotopy, encode displacement energy information and reflect rigidity phenomena in contact topology (Rizell et al., 2021).

7. Flexibility, h-Principle, and Limits of Invariants

The $h$-principle for loose Legendrians in high dimensions implies that for loose Legendrians (those containing a "loose chart" or with overtwisted complements), pseudo-holomorphic invariants such as LCH become trivial and fail to distinguish Legendrian isotopy classes. In such settings, classification reverts to smooth isotopy and formal invariants (almost complex framings) (Murphy, 2012, Ekholm, 2015). In dimension $3$, the corresponding statement is false, and Legendrian isotopy invariants retain their discriminatory power (Cavallo, 2017).

Invariant Type	Example Construction	Scope of Invariance
Chekanov–Eliashberg DGA/LCH	Holomorphic disks	All dimensions, standard (Y, ξ)
Sheaf Category $Sh(M,\Lambda^\infty)$	Singular support	Cosphere bundles, with Weinstein
GL-rack, combinatorial racks	Front diagrams	Knots/links in $(\mathbb{R}^3)$
Generating family spectra	Morse-theoretic	$J^1B$, with generating family
Page crossing number $M\mathcal{P}$	Open book decompositions	Legendrian surfaces in $5$-manifolds
Invariant distances/barcodes	Persistence modules	Isotopy class or universal covers

Research on Legendrian isotopy invariants thus intertwines algebraic, categorical, topological, and quantitative perspectives, reflecting the rich interplay between rigidity and flexibility in contact topology, with new invariants and generalizations continuing to emerge (Arlove, 24 Jul 2025, Ng, 2023, Tanaka et al., 2 Aug 2024, Zhou, 2018, Rizell et al., 2021, Arikan et al., 2021, Karmakar et al., 2023, Henry et al., 2014, Licata et al., 2010, Murphy, 2012, Ekholm, 2015, Cavallo, 2017).