Legendrian Satellite Operations

• Legendrian satellite operations are defined as constructions that embed a pattern Legendrian into a companion's neighborhood to produce new Legendrian submanifolds.
• They compute classical invariants through explicit formulas involving winding numbers, Thurston–Bennequin, and rotation numbers, crucial for classification.
• These operations extend to diverse applications including Lagrangian cobordisms, DGA invariants, and parametric families, bridging contact topology with algebraic methods.

Legendrian satellite operations are constructions in contact topology and knot theory that produce new Legendrian submanifolds by embedding a pattern Legendrian (typically in a solid torus or the 1-jet space of the circle) into a standard neighborhood of a companion Legendrian knot or link in a contact manifold, most commonly the standard contact $S^3$ or $\mathbb{R}^3$. These operations serve as a Legendrian analogue of classical knot satellite constructions, underpinning classification theory, the paper of Lagrangian cobordisms, contact invariants, and the interaction between smooth and contact-topological knot invariants.

1. Frameworks and Principle Constructions

Legendrian satellite operations are defined by embedding a pattern Legendrian $Q$ (in $(V, \xi_V)$, often $V = D^2 \times S^1$) into a neighborhood of a Legendrian knot $L \subset (S^3, \xi_{std})$, via a contactomorphism that identifies the product framing on $V$ with the contact framing of $L$ (Etnyre et al., 2016). The resulting Legendrian knot, denoted $Q(L)$, constitutes the Legendrian satellite of $L$ by $Q$.

These operations generalize classical cabling and connected sum constructions and are parameterized by winding numbers of the pattern, the Thurston–Bennequin (tb) and rotation number (rot) invariants, as well as the pattern's combinatorial type. The pattern may be a braid (yielding the braided satellite operation), a Whitehead pattern, or more intricate tangle or graph Legendrians.

Satellite operations extend naturally to parametric families of Legendrians, i.e., to maps between higher homotopy groups of spaces of Legendrian embeddings, as shown by the construction of parametric satellites and connected-sums at the homotopy level (Fernández et al., 29 May 2024). In this context, the operation gives rise to new invariants and facilitates the paper of the topology of the space of Legendrian links.

2. Invariants and Classification via Satellite Operations

The key classical invariants of Legendrian satellites are governed by explicit formulas. Given a pattern $Q$ of winding number $n$ with relative invariants $tb_V(Q)$ and $rot_V(Q)$, and a companion $L$ with $tb(L)$ and $rot(L)$, the invariants of the satellite are (Etnyre et al., 2016):

$tb(Q(L)) = n^2 \cdot tb(L) + tb_V(Q)$

$rot(Q(L)) = n \cdot rot(L) + rot_V(Q)$

For braided patterns, every Legendrian braid in the solid torus is isotopic to a cyclic concatenation of canonical building blocks (X, S, Z), classified via the relative Thurston–Bennequin and rotation numbers. For Whitehead patterns of winding zero, a rich classification emerges depending on the number and parity of half-twists $m$; for $m<0$ odd, there are $|m|+1$ maximal representatives, while for $m>0$ odd, only two maximal representatives exist but with distinct rotation numbers (Etnyre et al., 2016).

Under the assumption that the companion knot type is Legendrian simple and uniformly thick (all solid tori representing the knot isotoped inside a standard neighborhood of a maximal-tb representative), the Legendrian satellite operation induces a one-to-one correspondence between pairs (pattern, companion) modulo a stabilization equivalence and Legendrian isotopy classes of the satellite (Etnyre et al., 2016). For example:

$$Sat': (Leg_V(\Delta^{-t}P) \times Leg(K; t))/\sim \rightarrow Leg(P(K)), \quad (Q, L) \mapsto Q(L)$$

This bijection forms the technical basis for classifying Legendrian satellites and generalizes structure theorems for connected sums and cables (Etnyre et al., 2016).

3. Legendrian Satellites in the Context of Lagrangian Cobordisms

Legendrian satellite operations interact intimately with Lagrangian cobordism theory. Given decomposable Lagrangian concordances (cobordisms constructed from elementary moves such as Legendrian isotopy, 0-handles, and 1-handles), satellite operations lift these to corresponding cobordisms between satellites, provided appropriate twist corrections are made to compensate for the genus or topology of the underlying cobordism (Liu et al., 2017, Guadagni et al., 2021). The central geometric mechanism uses a Weinstein neighborhood model and push-off procedures to insert a tangle cobordism into a tubular neighborhood of the knot cobordism, yielding an exact Lagrangian cobordism between the corresponding satellites.

A fundamental result is that decomposable cobordisms between knots induce decomposable cobordisms between their satellites, modulo extra full twists related to the genus of the cobordism (Guadagni et al., 2021). Twist functions serve as combinatorial bookkeeping devices for tracking the interplay between handle moves and pattern twists. In formulas:

$$\{\Lambda_-\}^{(2g(L)+1)\Pi_-} \preceq \{\Lambda_+\}^{\Delta \Pi_+}$$

where extra twists are inserted according to the genus $g(L)$.

Legendrian satellite operations are also central to the theory of non-regular and non-decomposable Lagrangian concordances. By combining sufficient stabilizations, satellite operations such as the Legendrian Whitehead double (denoted $\Sigma(\Lambda, W_0)$), and inversion constructions, the existence of concordances in both directions between distinct fillable Legendrian knots can be realized, where one concordance is non-regular (Rizell et al., 16 Sep 2025). This demonstrates that the relation on Lagrangian concordances among fillable Legendrians is not antisymmetric.

4. Algebraic and Combinatorial Perspectives: DGAs and Moduli Spaces

Legendrian satellite operations are governed by powerful algebraic invariants. The Chekanov–Eliashberg differential graded algebra (DGA), combinatorially defined via crossings and vertex data, admits natural push-out diagrams under satellite operations such as tangle replacement (An et al., 2018). Specifically, the DGA of a Legendrian graph after a vertex is replaced by a tangle pattern is given by a push-out along the DGA of the vertex:

DGA $I_v$	DGA $A_T$	DGA $A_{L\langle T}$
Local vertex model	Pattern DGA	Satellite DGA

This formalism ensures that Legendrian contact homology and related algebraic structures "lift" appropriately under satellite operations.

Further, Legendrian weaves and satellite operations in higher dimensions are encoded via N-graph calculus, permitting explicit computation of microlocal sheaf moduli spaces and their point counts over finite fields (Casals et al., 2020). These moduli spaces provide analytic and combinatorial invariants of Legendrian satellites, and operations such as Legendrian mutation induce transformations captured by cluster algebra formulas on moduli coordinates. Both the algebraic and diagrammatic frameworks demonstrate that satellite operations interact compatibly with the structure of Legendrian invariants.

5. Extensions: Higher Homotopy, Lavrentiev Links, and Disconnected Legendrians

Legendrian satellite operations are not restricted to individual knots but extend to families and higher homotopy. Parametric satellite constructions operate on n-spheres of Legendrian embeddings, inducing group homomorphisms on higher homotopy groups of the embedding space (Fernández et al., 29 May 2024):

$$Sat(l)_*^n: \pi_n(\mathcal{L}, \gamma) \to \pi_n(\mathcal{L}, Sat(l)(\gamma))$$

This enables the construction of infinite families of loops with nontrivial Legendrian contact homology monodromy, enriching the space of Legendrian knots beyond the level of isotopy classifications.

Legendrian satellite operations extend to non-smooth categories such as Lavrentiev links—rectifiable curves with vanishing Riemann–Stieltjes integrals of the contact form on every subarc (Prasolov, 20 Apr 2024). The smoothing theorem guarantees that satellite operations defined on smooth Legendrians canonically extend to these broader classes under Legendrian isotopy.

Disconnected Legendrian submanifolds require a generalized understanding of augmentation homotopy and bilinearized Legendrian contact homology geography (Strakoš, 2022). The key criterion for DGA-homotopy under satellite operations is that augmentations $\epsilon_1 \sim \epsilon_2$ if and only if the fundamental class $[\Lambda]$ lies in the image of the top-degree duality map $\tau_n$, a criterion conducive to classifying satellites of links.

6. Summary and Technical Impact

Legendrian satellite operations provide a systematic and highly structured mechanism for constructing new Legendrian knots, links, and higher-dimensional submanifolds in contact manifolds, facilitating classification results, the paper of Lagrangian cobordisms, and the transfer of smooth knot invariants into the Legendrian field. Key advances include the explicit computation of classical and algebraic invariants under satellite operations, the lifting of cobordisms with precise twist corrections, the compatibility with advanced moduli and sheaf-theoretic invariants (including mutation and cluster phenomena), and the expansion to parametric and non-smooth settings.

As a central tool in modern contact topology, Legendrian satellite operations unify combinatorial, algebraic, and geometric approaches, serving as a bridge between Legendrian knot theory, symplectic geometry, and topological invariants. Their paper continues to produce new families of knots and links with subtle Legendrian and symplectic properties and reveals the intricate algebraic and topological structures underlying the mapping from patterns and companions to satellites.