Symmetric Legendrian Weaves

• Symmetric Legendrian weaves are Legendrian surfaces or spatial graphs in a contact manifold exhibiting explicit symmetry through combinatorial, geometric, and algebraic frameworks.
• They utilize weave calculus, cluster algebra combinatorics, and microlocal sheaf theory to establish invariant properties and facilitate Legendrian mutation and cluster structure analysis.
• Applications include the classification of exact Lagrangian fillings, analysis of augmentation varieties, and bridging concepts in topology, algebra, and geometry.

A symmetric Legendrian weave is a Legendrian surface or spatial graph embedded in a contact manifold whose combinatorial, geometric, and algebraic structure displays explicit symmetry—typically under a finite group of automorphisms such as a cyclic or dihedral group. These objects are studied through a combination of “weave calculus,” cluster algebra combinatorics, microlocal sheaf theory, and contact/symplectic topology. Symmetric Legendrian weaves serve as a bridge between topology, algebra, and geometry, encoding the symmetry and incidence data both in the front projections of Legendrian surfaces and in their moduli of microlocal sheaves, with direct implications for exact Lagrangian fillings, augmentation varieties, and cluster varieties.

1. Geometric and Combinatorial Construction of Symmetric Legendrian Weaves

Symmetric Legendrian weaves are constructed via combinatorial models such as colored planar graphs (N-graphs), trivalent graphs, or plabic graphs—subject to symmetry constraints.

• In the context of Legendrian torus links or their higher-dimensional twist-spun analogues, symmetric Legendrian weaves are often encoded by maximally symmetric objects. For a Legendrian link $\Lambda(k, n-k)$ fixed by a rotation $\psi$ of order $d = \frac{n}{\gcd(n,\ell)}$, a symmetric Legendrian weave arises from a maximal weakly separated collection $D \subset \binom{[n]}{k}$ satisfying $I +_n \ell \in D$ for all $I$. The dual plabic graph $G_D$ inherits the cyclic symmetry (rotation by $2\pi\ell/n$), which transfers to the corresponding weave via iterative T-shift procedures (Chen et al., 23 Sep 2025).
• In the planar spatial graph context, e.g. the $\Theta$-graph, an infinite family of symmetric Legendrian weaves arises by varying twist parameters while maintaining the overall symmetry of the graph and its embedding (Lambert-Cole et al., 2016).
• Geometrically, symmetric Legendrian weaves can be visualized as “multistrand” braided networks or surfaces in $J^1(\Sigma)$ (the 1-jet bundle), where the singular front projects to a colored, locally symmetric graph. Constructing such weaves involves local replacement rules (“moves”) that are compatible with the ambient symmetry (e.g., rotations or reflections) (Casals et al., 2020).

2. Symmetry, Weave Calculus, and Legendrian Mutation

Symmetry is implemented both at the combinatorial and topological level:

• The N-graph calculus provides a rigorous diagrammatic system in which colored edges and symmetry-invariant moves (e.g., the candy twist, push-throughs, flops) encode contact geometric operations and Legendrian isotopy while respecting group symmetry. Each local move can be tracked combinatorially to ensure that the symmetric pattern is preserved (Casals et al., 2020).
• Legendrian mutation is interpreted as a local alteration of the weave along cycles (I-cycles, Y-cycles), and in the symmetric context, only mutations that commute or intertwine suitably with the symmetry are admissible. Performing such mutations realizes explicit quiver mutations at the level of cluster algebra seeds, and the mutation-compatible cluster structures on moduli spaces respect these symmetries (Hughes, 2021, Hughes et al., 23 May 2025).
• For twist-spun or looped Legendrian surfaces, the underlying combinatorial symmetry (e.g., via cyclic action on subsets or rotation of plabic graphs) manifests as invariance under the mapping torus construction (Chen et al., 23 Sep 2025).

3. Cluster Algebra Structures and Foldings

Symmetric Legendrian weaves carry a naturally compatible cluster algebra structure on their sheaf-theoretic moduli:

• The sheaf moduli $\mathcal{M}_1(\lambda)$ associated to a Legendrian link $\lambda$ (e.g., the moduli of microlocal rank-1 constructible sheaves with singular support in $\Lambda$) is frequently a cluster variety. The underlying combinatorics—intersection quivers of the weave—yield cluster seeds. Each symmetric filling or weave corresponds to a cluster chart (toric coordinate chart) (Hughes et al., 23 May 2025).
• For symmetric Legendrian weaves built via twist spinning, the cluster algebra structure is globally foldable under the group action (e.g., the cyclic group generated by a rotation). The moduli for the twist-spun surface, e.g., $\mathcal{M}_1(\Sigma_\varphi(\lambda))$, is then isomorphic to the $G$-invariant subset of the original moduli, up to a $\mathbb{C}^*$ factor:

$$\mathcal{M}_1(\Sigma_\varphi(\lambda)) \cong \mathcal{M}_1(\lambda)^G \times \mathbb{C}^*,$$

where $G$ is the symmetry group (Hughes et al., 23 May 2025).

• The existence of these symmetric cluster structures allows for a “mutation distance” invariant to be defined: the minimal number of cluster mutations needed to relate two symmetric fillings. This is an isotopy invariant for the corresponding Legendrian doubles or twist spuns.
• The occurrence and count of symmetric Legendrian weaves are frequently controlled by combinatorial parameters (e.g., Catalan numbers, maximal weakly separated collections, or foldability conditions dictated by group actions) (Hughes, 2021, Chen et al., 23 Sep 2025).

4. Exact Lagrangian Fillings and Floer-theoretic Classification

Symmetric Legendrian weaves play a central role in the construction and classification of exact Lagrangian fillings.

• For many Legendrian links (especially positive braid closures), both decomposable fillings (via pinching cobordism sequences) and weave fillings (via projections of symmetric Legendrian weaves) are shown to be Hamiltonian isotopic and hence yield the same classes up to isotopy (Hughes, 2021).
• The correspondence between combinatorial invariants (e.g., 312-avoiding permutations or triangulations) and Lagrangian fillings is made explicit via bijections, such as the “clip-sequence” bijection between permutations and polygon triangulations, which is compatible with the action of loop symmetries (e.g., Kalman loop).
• The size of the orbit of a filling under such a Legendrian loop is governed by the symmetry of the underlying combinatorial data—full orbit if the triangulation is non-invariant under rotation; smaller orbits if there is rotational symmetry of order greater than 1 (Hughes, 2021).
• Floer-theoretic computations (using the Chekanov–Eliashberg DGA, augmentations, and recursive $\Delta$-functions on the augmentation variety) confirm the combinatorial and symmetry-based predictions, with identities such as Euler’s continuant formula providing key algebraic relationships.

5. Augmentation Varieties, Weave Decompositions, and Correspondences

Augmentation varieties present another categorical and algebraic manifestation of symmetric Legendrian weaves:

• The augmentation variety $\operatorname{Aug}(\lambda)$ of a Legendrian link, equipped with a cluster structure, can be decomposed using normal rulings, weaves, or Morse complex sequences (MCS). These decompositions correspond under functorial equivalences, and each stratum (corresponding to a ruling or right inductive weave) can be explicitly described as a product of tori and affine spaces (Asplund et al., 27 Aug 2025).
• Weaves induce algebraic correspondences on braid varieties and, after appropriate descent (modding out by torus actions corresponding to marked points), on augmentation varieties. These correspondences—compatible with symmetries—provide powerful tools for constructing and distinguishing Legendrian fillings and for relating structures in contact topology to those in cluster algebra and algebraic geometry (Casals et al., 2020).
• Cycle deletions and other operations, when applied with care respecting symmetry, allow for the construction of further decompositions, though some limitations and subtleties arise, especially regarding the preservation of right inductivity and ruling equivalence (Asplund et al., 27 Aug 2025).

6. Obstructions, Counting, and Combinatorial Criteria

Symmetry constraints both enable and obstruct the existence of symmetric Legendrian weaves and their fillings:

• Necessary and sufficient combinatorial criteria (e.g., $k \equiv -1,\,0,\,1 \pmod{d}$ for the existence of symmetric weakly separated collections of $k$-subsets) characterize when rotationally symmetric fillings of twist-spun Legendrian torus links exist (Chen et al., 23 Sep 2025).
• Obstructions to exact Lagrangian fillings manifest at the moduli level via the nonexistence of rational fixed points for certain group actions on Grassmannians, with explicit representation-theoretic and number-theoretic criteria used to prove non-fillability in certain cases (Hughes et al., 23 May 2025).
• The enumeration of symmetric Legendrian weaves and their fillings is often controlled by well-known combinatorial sequences such as the Catalan numbers, and by the structure of cluster exchange graphs under group actions (Hughes, 2021, Hughes et al., 23 May 2025).

7. Future Directions and Broader Connections

Research on symmetric Legendrian weaves interfaces with advanced topics in topology, representation theory, and algebraic geometry:

• The framework enables construction of Legendrian surfaces with prescribed symmetry groups (e.g., realizing any finite group as a subfactor in Lagrangian concordance monoids), explicit realization of cluster moduli, and exploration of Soergel bimodule and categorification connections (Casals et al., 2020).
• Comparison theorems establish the equivalence of different filling constructions (e.g., via plabic conjugation, Reeb pinching sequences, or weaves), and demonstrate that weaves generalize previous combinatorial methods while admitting strictly more classes (Casals et al., 2022).
• Further paper pursues higher rank sheaf invariants, derived structures for Legendrians with nontrivial Maslov class, and connections to nonabelian Hodge theory and partial compactifications (brick manifolds, Deodhar decompositions) (Casals et al., 2021, Asplund et al., 27 Aug 2025).
• The symmetry-driven combinatorics of symmetric Legendrian weaves continue to inform the interplay between topological invariants, cluster structures, and the geometry of moduli spaces, with applications to both mathematical physics (knot homologies, field theories) and low-dimensional topology.