[Paper Review] Legendrian Products

Peter Lambert‐Cole|arXiv (Cornell University)|Jan 16, 2013

Geometric and Algebraic Topology5 references4 citations

TL;DR

This paper introduces Legendrian products—a construction that combines two Legendrian submanifolds in contact manifolds of the form $P \times \mathbb{R}$ to produce new Legendrian submanifolds in $P \times Q \times \mathbb{R}$, with explicit formulas for their Thurston-Bennequin invariant and Maslov class. The key result shows that the isotopy class of the product depends on the specific embedding of the factors, not just their Legendrian isotopy class, leading to infinitely many non-isotopic products from a single isotopy class.

ABSTRACT

This paper introduces two constructions of Legendrian submanifolds, called the Legendrian product and spinning, and computes their classical invariants, the Thurston-Bennequin invariant and the Maslov class, in R^{2n+1}. These constructions take two Legendrians K,L and returns a product K x L, they generalize other previous constructions in contact topology, such as frontspinning and hypercube tori, and are equivalent in R^{2n+1}. Interestingly, this construction relies upon the explicit embeddings of K,L and not their Legendrian isotopy classes.

Motivation & Objective

• To develop a new construction of Legendrian submanifolds via products in $P \times \mathbb{R}$-type contact manifolds.
• To compute classical invariants—Thurston-Bennequin invariant and Maslov class—for these products.
• To demonstrate that the Legendrian isotopy class of the product depends on the specific embedding of the factors, not just their isotopy class.
• To generalize and unify existing constructions such as frontspinning and hypercube tori under a single framework.
• To provide explicit formulas for invariants in $\mathbb{R}^{2n+1}$, enabling concrete computations and new examples.

Proposed method

• Define the Legendrian product $K \times L$ as the lift of the Lagrangian product $\bar{K} \times \bar{L}$ in $P \times Q$, where $\bar{K}, \bar{L}$ are the Lagrangian projections of $K, L$.
• Ensure the product is embedded by requiring disjoint Reeb chord actions $\{Z(a_i)\}, \{Z(b_j)\}$; otherwise, it is immersed.
• Use the Reeb chord actions and signs $\sigma(a_i), \sigma(b_j)$ to compute linking numbers and classical invariants.
• Derive a formula for the Thurston-Bennequin invariant involving $tb(K), tb(L), \chi(T^*K), \chi(T^*L)$, and a sign-dependent sum over chord pairs.
• Introduce a sign function $\tau(a_i, b_j)$ that depends on the relative sizes of Reeb chord actions and the parities of the dimensions $n, m$ of the ambient spaces.
• Apply Stokes’s theorem to enforce area identities on faces of Lagrangian projections, constraining possible Reeb chord actions.

Experimental results

Research questions

• RQ1 Can a product construction of Legendrian submanifolds be defined in $P \times \mathbb{R}$-type contact manifolds that generalizes known constructions like frontspinning and hypercube tori?
• RQ2 How do the classical invariants—Thurston-Bennequin number and Maslov class—behave under such a product construction?
• RQ3 Does the Legendrian isotopy class of the product depend on the specific embedding of the factors, or only on their isotopy classes?
• RQ4 Can explicit formulas for the invariants be derived in $\mathbb{R}^{2n+1}$, and what do they reveal about the geometry of the product?
• RQ5 What is the range of possible Thurston-Bennequin invariants for products of three Legendrian knots, and how do they vary with chord actions and signs?

Key findings

• The Thurston-Bennequin invariant of $K \times L$ in $\mathbb{R}^{2n+1} \times \mathbb{R}^{2m+1}$ is given by a formula involving $tb(K)$, $tb(L)$, Euler characteristics of their unit cotangent bundles, and a sum over Reeb chord pairs with sign and action-dependent weights.
• The product construction depends on the specific embedding of the Legendrian factors, not just their isotopy classes, leading to infinitely many non-Legendrian-isotopic products from a single isotopy class.
• For three Legendrian knots $K_1, K_2, K_3$, the Thurston-Bennequin invariant of $K_1 \times K_2 \times K_3$ is computed as a sum over triples of Reeb chords with a function $\tau(a_i, b_j, c_k)$ that is 2 if the triangle inequality holds and 0 otherwise.
• The minimum possible $tb$ for a triple product of the given knot types is $-28$, and the maximum is $24$, with values in between achievable by adjusting Reeb chord actions.
• The area identity from Stokes’s theorem constrains Reeb chord actions: for each face in the Lagrangian projection, the sum of positive chord actions minus negative ones must equal the face area, which must be positive.
• By choosing chord actions appropriately, one can realize a wide range of $tb$ values—e.g., $tb = -28$ or $tb = 24$—demonstrating the richness of the construction in generating diverse invariants.