Weinstein Handlebody Diagrams

• Weinstein handlebody diagrams are graphical encodings of handle decompositions of Weinstein manifolds, particularly using Legendrian front projections in 4D.
• They facilitate explicit computation of topological and symplectic invariants, bridging Morse theory, algebra, and combinatorial methods.
• Their algorithmic construction and manipulation through Legendrian moves enable classification of symplectic structures and detection of flexibility or rigidity.

Weinstein handlebody diagrams are graphical encodings of the handle decompositions that define Weinstein manifolds, especially in dimension four, using the front projections of Legendrian attaching spheres in contact boundaries. These diagrams play a foundational role in symplectic topology, allowing explicit visualization, manipulation, and computation within exact symplectic and contact settings, and they form a bridge between Morse-theoretic, algebraic, and combinatorial perspectives on Weinstein domains.

1. Structure of Weinstein Handlebodies and Their Diagrams

A Weinstein manifold $(W,\omega,Z,\phi)$, with $\omega = d\lambda$, Liouville vector field $Z$, and gradient-like Morse function $\phi$, admits a decomposition by handles of index $\leq n$ in $2n$ dimensions. In dimension four ($n=2$), the decomposition uses a unique $0$-handle, a number of $1$-handles (represented diagrammatically by pairs of balls or dotted circles), and a number of $2$-handles attached along Legendrian knots in the contact boundary of the sublevel sets. The Legendrian attaching data is crucial, as Weinstein handle attachments are determined by contact-geometric framing: each $2$-handle is attached with smooth framing $tb(\Lambda)-1$, where $tb$ is the Thurston–Bennequin invariant of the Legendrian knot $\Lambda$ in the contact boundary.

Front projection diagrams, typically in Gompf standard form, organize this data efficiently. Legendrian knots are depicted so that their projection into the $xz$-plane records their contact and topological type, enabling explicit calculation of invariants such as $tb$ and rotation number. The resulting diagrams can be manipulated via a suite of Legendrian and Kirby moves (Legendrian Reidemeister I-III, Gompf moves 4-6, handle slides, and cancellations) to produce symplectically equivalent presentations (Acu et al., 2020, Acu et al., 2020, Ozbagci, 2018).

2. Algorithmic Construction and Satellite Realization

A central theme in modern Weinstein handlebody theory is the explicit, algorithmic production of handlebody diagrams corresponding to, for instance, complements of smoothed toric divisors, cotangent bundles, Milnor fibers, and moduli spaces. The general algorithm, as articulated in (Acu et al., 2020, Acu et al., 2020), is as follows:

• Step 0: Begin with the Gompf diagram for the disk cotangent bundle $D^*F$ of a closed surface $F$, including the $0$-handle, $2g$ $1$-handles (for genus $g$), and a single $2$-handle along a standard Legendrian.
• Step 1: Identify curves $\gamma_i$ on $F$ (e.g., conormal lifts of nontrivial slopes, vanishing cycles, or Stokes curves).
• Step 2: Realize these curves as Legendrian knots in the contact boundary using the "satellite construction," which involves front projections in $J^1(S^1)$ (the $S^1$-jet space model), adjusted to account for the $1$-handle layout.
• Step 3: Attach $2$-handles along these Legendrians with framing $tb-1$.
• Step 4: Simplify the diagram using topological and Legendrian moves to extract a minimal or standard form for the resulting Weinstein manifold.

Explicit application of this strategy yields handlebody diagrams for a wide range of examples, including the complements of toric divisors (Acu et al., 2020, Acu et al., 2020), Milnor fibers with $T_{p,q,r}$ singularities (Capovilla-Searle, 2022), Painlevé moduli spaces (Beimler et al., 21 Nov 2025), and more.

3. Lefschetz Fibrations, Legendrian Correspondence, and Multisections

Many Weinstein manifolds are presented using (positive allowable) Lefschetz fibrations, where vanishing cycles in fibered surfaces are mapped to Legendrian attaching data for $2$-handles. The translation from fibration-theoretic to handlebody-theoretic language leverages systematic "affine dictionaries," as in (Casals et al., 2016) and (Islambouli et al., 2023).

• Each vanishing cycle gives rise to a Legendrian link in the contact boundary.
• The entire Weinstein structure is then encoded as attachment of $2$-handles along these links.
• Multisection diagrams with divides extend the approach by embedding the entire handle structure in a higher-genus Heegaard-splitting framework, facilitating monodromy substitutions and symplectic surgeries (Islambouli et al., 2023).

This correspondence is particularly powerful for explicit detection of properties such as flexibility, the existence of exact Lagrangian submanifolds, or the calculation of symplectic invariants.

4. Computation of Invariants via Handle Diagrams

Weinstein handlebody diagrams encode all data necessary to compute topological (e.g., fundamental group, homology, intersection forms) and symplectic invariants (e.g., symplectic and wrapped Fukaya categories, symplectic homology). The algebraic readings are summarized as follows (Lazarev, 2019, Acu et al., 2020):

• Homology and Intersection Forms: Extracted from the linking matrix determined by the attaching link in the diagram, with self-linking given by $tb-1$ for each component.
• Symplectic Homology: Invariants such as $SH_*(X)$ can be computed from the Legendrian contact homology of the boundary, with explicit non-vanishing criteria based on the existence of graded representations of the Chekanov–Eliashberg DGA of a sublink (Capovilla-Searle, 2022, Acu et al., 2020).
• Wrapped Fukaya Category: The equivalence between geometric intersections of handles and algebraic relations induces surjective maps from singular cohomology to the Grothendieck group of the Fukaya category, enabling detection of derived Morita equivalence and symplectic flexibility or rigidity (Lazarev, 2019).

These features underpin the use of Weinstein handlebody diagrams as both combinatorial and algebraic tools.

5. Applications: Symplectic Topology and Lagrangian Constructions

Weinstein handlebody diagrams provide a direct and flexible means to construct exact Lagrangian submanifolds, exhibit exotic presentations, and distinguish non-isomorphic Weinstein structures. For example:

• The handlebody for $T_{p,q,r}$ singularities enables the explicit construction of infinitely many distinct, pairwise non-Hamiltonian isotopic exact Maslov-zero Lagrangian tori in Milnor fibers (Capovilla-Searle, 2022).
• Algorithmic use of augmentations and Legendrian invariants produces further families of exact Lagrangians in other symplectic four-manifolds, including complements of toric divisors and plumbed cotangent bundles (Acu et al., 2020, Acu et al., 2020).
• Exotic presentations and flexible/rigid dichotomies are accessible through manipulation of the handle diagram, using known correspondences with the wrapped Fukaya category and Chekanov–Eliashberg DGA isomorphism types (Lazarev, 2019, Casals et al., 2016).

The naturality and explicitness of the diagrams permit effective deployment in both construction and classification problems throughout symplectic topology.

6. Moduli Spaces, Degenerations, and New Examples

Recent advances encompass the construction of Weinstein handlebody diagrams for moduli spaces associated to classically integrable systems (e.g., Painlevé Betti moduli), where handle attachments are derived from Stokes data and singularity resolutions (Beimler et al., 21 Nov 2025). This yields:

• Universal combinatorial recipes: Begin from the cotangent bundle of a punctured surface, attach $2$-handles along Legendrian lifts of cooriented curves determined by Stokes or vanishing cycle data, and simplify using Gompf moves to extract minimal models.
• Control over boundary topology and symplectic invariants: For example, Painlevé VI Betti surfaces manifest as Weinstein manifolds with explicit $1$- and $2$-handle configuration directly related to the moduli space stratification.
• Experimental access to degenerations, symplectic surgeries, and wall-crossing phenomena realized through monodromy substitutions at the diagrammatic level (Islambouli et al., 2023).

This expansion demonstrates the versatility of Weinstein handlebody diagrams for both classical and new types of symplectic manifolds.

7. Summary Table: Structure and Key Features

Aspect	Weinstein Handlebody Diagrams	References
Initial Data	Legendrian front projections, 0/1/2-handle specification, contact framings	(Ozbagci, 2018, Acu et al., 2020, Acu et al., 2020)
Algorithmic Recipes	Satellite construction, explicit computation from curves on surfaces	(Acu et al., 2020, Acu et al., 2020, Beimler et al., 21 Nov 2025)
Calculable Invariants	Homology, intersection form, symplectic/Legendrian invariants, Floer data	(Lazarev, 2019, Capovilla-Searle, 2022, Acu et al., 2020)
Diagram Moves	Legendrian Reidemeister, Gompf 4–6, handle slides/cancellations	(Acu et al., 2020, Acu et al., 2020, Ozbagci, 2018)
Flexibility Criteria	Chekanov algebra representations, non-vanishing symplectic homology, sublink augmentations	(Capovilla-Searle, 2022, Casals et al., 2016, Acu et al., 2020)

Explicit handlebody diagrams are thus central tools in the construction, computation, and classification of Weinstein manifolds, especially in dimension four, linking combinatorial, geometric, and algebraic methods in symplectic topology (Capovilla-Searle, 2022, Beimler et al., 21 Nov 2025, Islambouli et al., 2023).