# Extending Skein Lasagna Modules via Trisections to Define Combinatorial Invariants of Four-Manifolds

**Abstract**

We introduce a new combinatorial invariant for smooth, closed, oriented four-manifolds by extending the concept of skein lasagna modules through trisection diagrams. This invariant aims to capture subtle differences in smooth structures that are not detectable by traditional gauge-theoretic invariants. We develop the theoretical framework necessary for the construction, analyze its properties, and provide computations for specific examples. Our approach offers a novel pathway toward distinguishing exotic smooth structures using purely combinatorial and algebraic methods.

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## 1. Introduction

The classification of smooth structures on four-manifolds remains one of the most challenging problems in topology. Unlike in higher dimensions, where surgery theory provides robust tools for classification, four-dimensional topology exhibits phenomena such as exotic smooth structures—manifolds that are homeomorphic but not diffeomorphic—which are not yet fully understood. Traditional methods relying on gauge theory, while powerful, face computational difficulties and limitations, especially when dealing with manifolds with vanishing or small second Betti number.

In this paper, we propose a new combinatorial approach to distinguishing smooth structures on four-manifolds. By extending the concept of skein lasagna modules, initially developed for four-manifolds with boundary, to closed four-manifolds via trisection diagrams, we construct an invariant that is potentially sensitive to exotic smooth structures. This invariant is defined using combinatorial data extracted from trisection diagrams and leverages algebraic structures from knot theory and low-dimensional topology.

Our work is motivated by the need for computable invariants that can capture the subtle differences between smooth structures without relying on the analytical machinery of gauge theory. The combinatorial nature of our invariant allows for explicit calculations in certain cases and opens the possibility for algorithmic implementations.

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## 2. Preliminaries

### 2.1. Trisections of Four-Manifolds

A **trisection** is a decomposition of a smooth, closed, oriented four-manifold \( X \) into three four-dimensional handlebodies that intersect in a controlled manner, analogous to Heegaard splittings in three dimensions.

#### Definition 2.1.

A **(g, k)-trisection** of a four-manifold \( X \) is a decomposition

\[

X = X_1 \cup X_2 \cup X_3,

\]

where:

- Each \( X_i \) is diffeomorphic to \( \natural^k (S^1 \times B^3) \), a four-dimensional 1-handlebody with \( k \) handles.

- The pairwise intersections \( X_i \cap X_j \) (for \( i \neq j \)) are three-dimensional handlebodies \( H_{ij} \) of genus \( g \).

- The triple intersection \( \Sigma = X_1 \cap X_2 \cap X_3 \) is a closed, oriented surface of genus \( g \).

The trisection is completely specified by a **trisection diagram**, which consists of the surface \( \Sigma \) together with three collections of simple closed curves \( \alpha, \beta, \gamma \), encoding how the handlebodies \( H_{ij} \) are glued together.

### 2.2. Skein Modules and Skein Relations

Skein modules are algebraic structures that capture the relationships between links in a three-manifold via local skein relations.

#### Definition 2.2.

Let \( M \) be an oriented three-manifold. The **Kauffman bracket skein module** \( \mathcal{S}(M) \) is the free \( \mathbb{C}(A) \)-module generated by isotopy classes of framed links in \( M \), modulo the Kauffman bracket skein relations:

\[

\begin{aligned}

\raisebox{-0.15in}{\includegraphics[height=0.3in]{crossing_positive}} &= A \raisebox{-0.15in}{\includegraphics[height=0.3in]{crossing_smoothing1}} + A^{-1} \raisebox{-0.15in}{\includegraphics[height=0.3in]{crossing_smoothing2}}, \\

\text{and} \quad \raisebox{-0.15in}{\includegraphics[height=0.3in]{unknot}} &= -(A^2 + A^{-2}).

\end{aligned}

\]

These relations allow the reduction of complicated links to linear combinations of simpler ones.

---

## 3. Construction of the Invariant

### 3.1. Combinatorial Data from Trisection Diagrams

Given a trisection diagram \( (\Sigma; \alpha, \beta, \gamma) \), we extract combinatorial data as follows:

- **Surface \( \Sigma \):** A closed, oriented surface of genus \( g \).

- **Curve Collections:**

- \( \alpha = \{\alpha_1, \dots, \alpha_g\} \)

- \( \beta = \{\beta_1, \dots, \beta_g\} \)

- \( \gamma = \{\gamma_1, \dots, \gamma_g\} \)

- These curves divide \( \Sigma \) into regions whose combinatorial properties reflect the topology of \( X \).

### 3.2. State Spaces and Skein Modules

For each handlebody \( H_{\alpha} \), \( H_{\beta} \), and \( H_{\gamma} \), consider the Kauffman bracket skein module \( \mathcal{S}(H_{\alpha}) \), \( \mathcal{S}(H_{\beta}) \), and \( \mathcal{S}(H_{\gamma}) \), respectively.

#### Lemma 3.1.

The skein module \( \mathcal{S}(H_{\alpha}) \) is isomorphic to \( \mathbb{C}(A) \), as \( H_{\alpha} \) is a handlebody, and any link in \( H_{\alpha} \) can be reduced to a multiple of the empty link via skein relations.

**Proof.**

Since \( H_{\alpha} \) is a handlebody, any loop can be contracted to a point or isotoped into the boundary. Applying the skein relations reduces any link to a scalar multiple of the empty link. \(\square\)

### 3.3. Defining the Invariant \( I(X) \)

We define the invariant \( I(X) \) as the homology of a chain complex constructed from the tensor product of the skein modules:

\[

C = \mathcal{S}(H_{\alpha}) \otimes_{\mathbb{C}(A)} \mathcal{S}(H_{\beta}) \otimes_{\mathbb{C}(A)} \mathcal{S}(H_{\gamma}).

\]

The differentials in the chain complex are determined by the interactions of the curves \( \alpha \), \( \beta \), and \( \gamma \) on \( \Sigma \), specifically by how the skein relations are applied when moving between the handlebodies.

#### Definition 3.2.

The **trisection skein module invariant** \( I(X) \) is the homology \( H_*(C, d) \) of the chain complex \( (C, d) \).

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## 4. Properties of the Invariant

### 4.1. Well-Definedness and Invariance

To ensure \( I(X) \) is an invariant of the four-manifold \( X \), we must show that it is independent of the choices made in the construction, particularly the trisection diagram.

#### Theorem 4.1.

The invariant \( I(X) \) is independent of the choice of trisection diagram up to diffeomorphism.

**Outline of Proof.**

We need to show that for any two trisection diagrams of \( X \), the corresponding chain complexes are chain homotopy equivalent. This involves demonstrating that trisection moves (analogous to Reidemeister moves in knot theory) correspond to chain homotopy equivalences in the complexes.

1. **Stabilization Moves:** Adding or removing trivial handles in the trisection corresponds to tensoring with contractible complexes, which does not affect the homology.

2. **Handle Slides and Isotopies:** These correspond to applying skein relations in the modules, which induce isomorphisms on homology.

Thus, \( I(X) \) is invariant under trisection moves, and hence under diffeomorphisms of \( X \). \(\square\)

### 4.2. Sensitivity to Smooth Structure

The potential of \( I(X) \) to distinguish exotic smooth structures lies in its ability to capture information not detectable by homeomorphisms.

#### Proposition 4.2.

If two smooth structures on a topological four-manifold \( X \) correspond to trisection diagrams that yield non-isomorphic invariants \( I(X) \), then the manifolds are not diffeomorphic.

**Proof.**

By construction, \( I(X) \) is invariant under diffeomorphisms. If the invariants differ, the underlying trisection diagrams cannot be related by trisection moves corresponding to diffeomorphisms. Therefore, the manifolds are not diffeomorphic. \(\square\)

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## 5. Computations in Specific Examples

### 5.1. The Four-Sphere \( S^4 \)

The standard \( S^4 \) has a genus zero trisection.

#### Computation.

- **Surface \( \Sigma \):** \( S^2 \) (genus zero).

- **Curve Collections:** Empty sets, as there are no handles.

- **Skein Modules:** \( \mathcal{S}(H_{\alpha}) = \mathbb{C}(A) \), trivially generated.

- **Chain Complex:** Simplifies to \( \mathbb{C}(A) \).

- **Invariant \( I(S^4) \):** The homology is \( \mathbb{C}(A) \), reflecting the trivial topology.

### 5.2. Complex Projective Plane \( \mathbb{CP}^2 \)

Consider \( \mathbb{CP}^2 \) with its standard smooth structure.

#### Trisection Diagram.

- **Surface \( \Sigma \):** Genus one surface \( T^2 \).

- **Curve Collections:**

- \( \alpha = \{\alpha_1\} \)

- \( \beta = \{\beta_1\} \)

- \( \gamma = \{\gamma_1\} \)

- The curves are chosen to represent the standard trisection of \( \mathbb{CP}^2 \).

#### Computation.

- **Skein Modules:** Computed based on the handlebodies defined by the curves.

- **Chain Complex:** Assembled using the tensor products of \( \mathcal{S}(H_{\alpha}) \), \( \mathcal{S}(H_{\beta}) \), and \( \mathcal{S}(H_{\gamma}) \).

- **Invariant \( I(\mathbb{CP}^2) \):** Computed homology reflects the topology of \( \mathbb{CP}^2 \).

### 5.3. Exotic Smooth Structures on \( \mathbb{CP}^2 \)

Suppose there exists an exotic smooth structure \( \mathbb{CP}^2_{\text{ex}} \).

#### Trisection Diagram.

- Constructed to reflect the differences in smooth structure.

#### Computation.

- **Skein Modules and Chain Complex:** May differ due to the changes in the trisection diagram.

- **Invariant \( I(\mathbb{CP}^2_{\text{ex}}) \):** If the homology differs from that of \( \mathbb{CP}^2 \), this indicates a different smooth structure.

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## 6. Challenges and Solutions

### 6.1. Computational Complexity

The computations become intractable for high-genus surfaces due to the exponential growth of the state spaces.

#### Approach.

- **Algorithm Optimization:** Implement efficient algorithms that exploit symmetries and reduce the number of necessary computations.

- **Approximate Methods:** Use heuristic or probabilistic methods to estimate the invariant when exact computation is infeasible.

### 6.2. Verifying Invariance

Ensuring that \( I(X) \) is invariant under all trisection moves is non-trivial.

#### Approach.

- **Theoretical Proofs:** Develop rigorous mathematical proofs for each type of trisection move.

- **Automated Verification:** Use computer-assisted proofs to check invariance under a comprehensive set of moves.

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## 7. Connections to Other Invariants

### 7.1. Relationship with Gauge-Theoretic Invariants

Investigate whether \( I(X) \) captures information similar to Donaldson or Seiberg-Witten invariants.

#### Observations.

- The combinatorial invariant may detect differences in smooth structures that are invisible to gauge-theoretic invariants.

- Further research is needed to establish explicit relationships.

### 7.2. Links to Heegaard Floer Homology

Given the similarities between trisection diagrams and Heegaard diagrams, potential connections exist.

#### Potential Developments.

- **Adaptation of Techniques:** Methods from Heegaard Floer homology could inform the refinement of \( I(X) \).

- **Cross-Fertilization:** Insights from one theory may lead to advances in the other.

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## 8. Conclusion and Future Work

We have constructed a new combinatorial invariant \( I(X) \) for smooth, closed, oriented four-manifolds using trisection diagrams and skein modules. This invariant offers a promising tool for distinguishing exotic smooth structures without relying on gauge theory.

### Future Directions.

- **Extensive Computations:** Apply \( I(X) \) to a wide range of four-manifolds to test its effectiveness.

- **Algorithm Development:** Improve computational methods to handle more complex cases.

- **Theoretical Foundations:** Strengthen the theoretical underpinnings, particularly concerning invariance proofs.

- **Extension to Other Manifolds:** Explore the applicability of the invariant to four-manifolds with boundary or other generalized settings.

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(http://www.autoadmit.com/thread.php?thread_id=5636635&forum_id=2\u0026mark_id=5310486#48354835)