Skein-Valued Curve Counting

• Skein-valued curve counting is a framework that refines classical Gromov-Witten invariants by encoding holomorphic curve counts as elements of skein modules.
• It employs robust perturbation theory and polyfold models to ensure transversality and compactness of moduli spaces, achieving deformation-invariant counts.
• The approach connects open string enumerative invariants with knot polynomials, integrating mirror symmetry, gauge theories, and cluster transformations.

Skein-valued curve counting is a theory that organizes enumerative holomorphic curve invariants in symplectic and algebraic geometry as elements of skein modules, thereby providing a quantum-topological refinement of classical Gromov-Witten theory. In this framework, moduli spaces of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds are enumerated not merely by the number or cohomology class of the curves, but by the isotopy class of their boundaries within the skein algebra of the Lagrangian. This construction yields topological invariants, matches wall-crossing phenomena in curve enumeration to skein relations in quantum topology, and rigorously connects open string enumerative invariants to knot and link polynomials, notably fulfilling predictions from topological string theory such as the Ooguri-Vafa conjecture. The structure and invariance of these counts depend crucially on modern perturbation theory, polyfold models, and categorical tools, and have further implications in cluster transformations, quantum curves, and gauge-theoretic perspectives.

1. Geometric Framework and Moduli Spaces

Skein-valued curve counting begins with stable holomorphic maps $u\colon(\Sigma,\partial\Sigma)\to(X,L)$ where $X$ is a symplectic Calabi–Yau 3-fold $(c_1(X)=0)$ and $L\subset X$ is a Maslov-zero Lagrangian submanifold, necessarily orientable and spin (Ekholm et al., 2019). The moduli spaces $M_{g,h}(X,L;\beta)$ are composed of $J$-holomorphic curves (possibly disconnected, genus $g$, $h$ boundary components) mapping into $X$ with boundary on $L$ and homology class $\beta \in H_2(X,L)$. Through a combination of Gromov-compactness and ghost-bubble exclusion arguments, these moduli spaces admit compactifications and, with adequate generic perturbations, become transversely cut out such that their zero-dimensional components define finite counts (Ekholm et al., 2024).

2. Skein Modules and Algebraic Formalism

The essential algebraic object is the framed HOMFLYPT skein module $Sk(L)$, a free $\mathbb{Z}[a^{\pm1},z^{\pm1}]$-module generated by isotopy classes of framed, oriented links in $L$ modulo local skein relations:

• Crossing change: $L_+ - L_- = (q^{1/2}-q^{-1/2}) L_0$,
• Framing twist: $L_{↺} = a L$, where $q=z^2$ (Ekholm et al., 2019, Ekholm et al., 2020).

The boundary of a holomorphic curve $\partial u \subset L$ is associable to a skein class $[\partial u]\in Sk(L)$, with the skein relations encoding the topological changes under deformation. The completed skein module $\widehat{Sk}(L)$ allows for refinements by "energy" or winding number gradings for noncompact Lagrangians.

3. Wall-crossing, Codimension-One Degenerations, and Invariance

A central insight is the matching of moduli space boundary phenomena with skein algebra relations. Under generic one-parameter deformations of the almost complex structure, the only boundary phenomena for the moduli space are nodes of hyperbolic (locally $x^2-y^2=0$) or elliptic (locally $x^2+y^2=0$) type, and simple framing tangencies (Ekholm et al., 2019). These correspond precisely to:

• Hyperbolic node: $L_+ - L_- - z L_0 = 0$,
• Elliptic node: $a L_+ - a^{-1} L_- - z L_0 = 0$,
• Framing twist: $L_{↺} - aL = 0$.

The skein-valued partition function

$Z_L = \sum_{g,h,\beta}\sum_{u\in M_{g,h}(X,L;\beta)} w(u)\,z^{-\chi(u)}\,a^{u\OpenHopf L}\,[\partial u] \in \widehat{Sk}(L)$

is deformation invariant: changes in $J$, perturbation data, or bounding chains only introduce skein relations at wall-crossings, guaranteeing the count remains unchanged (Ekholm et al., 2019, Ekholm et al., 2024).

4. Analytic Foundations: Adequate Perturbation Schemes

Transversality and compactness of the relevant moduli spaces require an adequate perturbation scheme, constructed in the Hofer-Wysocki-Zehnder polyfold framework (Ekholm et al., 2024). Such a multi-section $\theta:Z\to W$ for the configuration space $(Z,W)$ must:

• Vanish on zero-symplectic-area ("ghost") components and be supported away from $L$,
• Forbid ghost-bubble contributions to bare counts,
• Ensure all regular bare solutions are transversely cut out, compact, and coherently oriented,
• Guarantee only codimension-one degenerations occur in one-parameter families (no ghost bubbling). These conditions yield skein-valued Gromov-Witten invariants satisfying deformation invariance, quantum gluing (HOMFLYPT skein relations), and disk-capping (multiplication by $Q-Q^{-1}$).

5. Quantum Curve Recursion and Mirror Symmetry

Curves with boundary on toric branes in $\mathbb{C}^3$ or more general toric Calabi-Yau 3-folds must satisfy skein-theoretic recursion relations, often corresponding to quantizations of mirror curves of the geometry (Ekholm et al., 2020, Hu et al., 8 Dec 2025). For a solid torus brane $L$, the generating function $\Psi_L$ in $Sk(L)$ obeys

$(1 - P_{1,0} + a_L \gamma P_{0,1})\Psi_L = 0,$

where $P_{m,n}$ corresponds to winding operators. The unique solution in the idempotent basis is given by the hook-content formula involving products over Young diagrams, matching the colored HOMFLYPT invariants.

For toric strips, the quantum mirror operator annihilates the skein-valued generating function, with closed-form solutions expressible via skein dilogarithms. These constructions establish a direct link between holomorphic curve enumeration, quantum curves, and skein-theoretical recursion (Hu et al., 8 Dec 2025).

6. Cluster Transformations and Wall-Crossing Extensions

The theory extends to skein-valued cluster transformations under Legendrian mutation or Lagrangian surgery. Under suitable geometric hypotheses, skein-valued curve counts transform via conjugation by skein dilogarithms, upgrading $q$-cluster mutations to HOMFLYPT skein level (Scharitzer et al., 2023). This matches wall-crossing formulas of Kontsevich-Soibelman type and yields identities such as skein pentagon relations, which are realized combinatorially in genus-two examples via necklace decompositions and mutation sequences.

These results provide a geometric underpinning to algebraic cluster structures, wall-crossing, and quantum dilogarithm identities appearing in knot theory, DT theory, and quantum character varieties.

7. Physical Interpretation: Gauge Theory and M-Theory Perspective

Gauge-theoretic frameworks embed skein modules into states of twisted four-dimensional $\mathcal{N}=4$ SYM theories, with skein-valued curve counts arising naturally in the context of 4d-6d correspondences, BPS state enumeration, and open topological strings (Pei, 22 Jan 2026). Holomorphic curves map to boundary states defined by Wilson lines; full invariance under deformation reflects the physical invariance under changes in background or brane data. The connection elucidates the enumerative meaning of knot polynomials, the role of Langlands duality in the structure of skein modules, and the physical realization of cluster transformations and quantum curves.

8. Refined Enumerative Invariants and Recursion Structures

The skein-valued formalism refines classical Gromov-Witten invariants by promoting counts to elements encoding full framed boundary data. It incorporates gluing identities, multi-cover relations, and crossing formulas corresponding to elliptic and hyperbolic nodes, exemplified by skein pentagon and cluster exchange relations (Nakamura, 2024). These features ensure compatibility with SFT gluing, BPS expansions, and offer a systematic framework for all-genus and multi-boundary enumerative invariants, extending open/closed duality and geometric recursion structures.

9. Applications and Computational Schemes

Skein-valued curve counts are foundational in computing open Gromov-Witten invariants for knot conormals, toric branes, and more general symplectic manifolds, reproducing knot polynomials (HOMFLYPT, colored invariants), topological vertex formulas, and quantum character varieties. Uniqueness results ensure the recursive operator equations fix partition functions entirely (Ekholm et al., 2024, Ekholm et al., 2024). The methods are compatible with computational schemes arising in quantum topology, cluster algebra, and wall-crossing, and yield explicit formulae for BPS partition functions, providing a bridge between algebraic, geometric, and physical approaches to enumerative geometry.

---

The skein-valued curve counting paradigm thus integrates moduli space analysis, quantum-topological algebra, and physical dualities, yielding deformation-invariant, skein-theoretic refinements of enumerative holomorphic curve invariants, with deep connections to knot theory, mirror symmetry, and gauge-theoretic structures (Ekholm et al., 2019, Ekholm et al., 2024, Ekholm et al., 2020, Scharitzer et al., 2023, Hu et al., 8 Dec 2025, Ekholm et al., 21 Oct 2025, Ekholm et al., 2024, Ekholm et al., 2024, Pei, 22 Jan 2026, Nakamura, 2024).