Maximal Page Crossing Numbers of Legendrian Surfaces in Closed Contact 5-Manifolds

Abstract: We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$-manifold $(M, ξ)$ which admits an "admissable" open book $(B, f)$ (supporting $ξ$) for $L$. We show that to any such $L$ and a fixed page $X$, one can assign an integer $M\mathcal{P}{X}(L)$, called "Relative Maximal Page Crossing Number of $L$ with respect to $X$", which is invariant under Legendrian isotopies of $L$. We also show that one can extend this to a page-free invariant, i.e., one can assign an integer $M\mathcal{P}{(B,f)}(L)$, called "Absolute Maximal Page Crossing Number of $L$ with respect to $(B, f)$", which is invariant under Legendrian isotopies of $L$. In particular, this new invariant distinguishes Legendrian surfaces in the standard five-sphere which can not be distinguished by Thurston-Bennequin invariant.

• The paper introduces new invariants, the Relative and Absolute Maximal Page Crossing Numbers, to distinguish Legendrian surfaces when classical invariants fail.
• The methodology leverages admissible open book decompositions and Stein handle diagram techniques to compute page crossing numbers via Thurston-Bennequin evaluations.
• The invariants, validated on examples like the Clifford torus, offer practical tools for classifying Legendrians in closed contact 5-manifolds.

Maximal Page Crossing Numbers of Legendrian Surfaces in Closed Contact 5-Manifolds

Introduction and Motivation

The study of Legendrian submanifolds within contact manifolds is central to higher-dimensional contact topology. While the Thurston-Bennequin invariant effectively distinguishes Legendrian knots in three dimensions, its discriminative power sharply decreases for Legendrian surfaces in contact 5-manifolds, as it often collapses to a purely topological invariant. This paper introduces new Legendrian isotopy invariants—the Relative Maximal Page Crossing Number $M\mathcal{P}_X(L)$ and the Absolute Maximal Page Crossing Number $M\mathcal{P}_{(B,f)}(L)$—tailored to closed orientable Legendrian surfaces $L$ in closed contact 5-manifolds $(M,\xi)$ supporting admissible open book decompositions. These invariants exploit the interplay between open book decompositions and the topology of Legendrian surfaces to yield computable, page- and binding-dependent data that are sensitive to Legendrian isotopy, even when classical invariants fail.

Definitions and Key Construction

Admissible Open Books and Page Crossing

For a closed contact 5-manifold $(M^5,\xi)$, an admissible open book $(B, f)$ is one where the binding $B$ and the Legendrian $L$ intersect transversely, and each page $X$ is a simply-connected Weinstein domain. The key insight is that the interaction of $L$ with the double $M\mathcal{P}_{(B,f)}(L)$0 (the union of dual pages) can, via this transversality, be encoded as a link in $M\mathcal{P}_{(B,f)}(L)$1.

Maximal Page Crossing Numbers

The Relative Maximal Page Crossing Number $M\mathcal{P}_{(B,f)}(L)$2 is defined as follows:

• For a fixed page $M\mathcal{P}_{(B,f)}(L)$3, consider all Legendrian representatives $M\mathcal{P}_{(B,f)}(L)$4 isotopic to $M\mathcal{P}_{(B,f)}(L)$5 that intersect $M\mathcal{P}_{(B,f)}(L)$6 transversely and essentially.
• For each such $M\mathcal{P}_{(B,f)}(L)$7, the intersection $M\mathcal{P}_{(B,f)}(L)$8 forms a link $M\mathcal{P}_{(B,f)}(L)$9 in $L$0, decomposed into components $L$1 (arcs in $L$2 and $L$3).
• Assign to each $L$4 the sum $L$5 using Stein handle diagram techniques.
• Sum over components to define the page crossing number $L$6.
• Take the maximum over all $L$7, i.e., $L$8.

The Absolute Maximal Page Crossing Number $L$9 is the value $(M,\xi)$0 for any page $(M,\xi)$1 in the admissible open book $(M,\xi)$2, proven to be independent of the page choice by compactness and invariance arguments.

Invariance and Well-Definedness

Both $(M,\xi)$3 and $(M,\xi)$4 are shown to be:

• Invariant under Legendrian isotopy: They do not change under regular or irregular Legendrian isotopies with respect to the double $(M,\xi)$5, as detailed in rigorous combinatorial and topological arguments.
• Well-defined: Zorn's lemma–based arguments and compactness ensure that there are maximal representatives, and the invariants are not subject to infinite increases or ambiguity.

Key numerical property: If link components in the intersection are homotopically trivial and arise as unknots, then their Thurston-Bennequin numbers are strictly negative, ensuring that maximality occurs for links with only nontrivial homology and minimal geometric intersection with the binding.

Comparison with Thurston-Bennequin Invariant

A notable feature of these invariants is their power to distinguish Legendrian surfaces in $(M,\xi)$6 that are not distinguished by the Thurston-Bennequin invariant. For Legendrian tori, for instance, the Thurston-Bennequin number is always zero due to topological constraints [EES5], whereas the maximal page crossing numbers can be nonzero and Legendrian isotopy sensitive.

Computability and Example

A concrete example in the paper considers the Clifford torus $(M,\xi)$7 in $(M,\xi)$8, using the standard open book decomposition with binding $(M,\xi)$9. The torus $(M^5,\xi)$0 is shown to intersect each page double along a pair of unlinked Legendrian unknots, each with Thurston-Bennequin number $(M^5,\xi)$1. This yields

$(M^5,\xi)$2

whereas $(M^5,\xi)$3, demonstrating the distinction and computability of the new invariant.

Theoretical and Practical Implications

Theoretical advances:

• The construction provides an isotopy invariant for higher-dimensional Legendrians that is far more readily computable than Legendrian contact DGA invariants, relying on classical diagrammatic moves in Stein handlebody diagrams.
• These invariants can distinguish Legendrian isotopy classes even when all known classical invariants coincide, increasing the resolution of the isotopy classification of Legendrian surfaces in high dimensions.
• The methodology leverages the Giroux correspondence, linking contact topology, open book decompositions, and Stein/Weinstein geometry, suggesting avenues for new relationships between classical invariants and open book–based invariants in contact topology.

Practical utility:

• Since the invariants only require manipulation of Stein diagrams and classical calculation of Thurston-Bennequin numbers, they are tractable in explicit constructions—a significant advantage compared to previously available invariants.
• The invariants provide a new tool for understanding Legendrian embeddings in the context of symplectic and contact topology, with potential applications to symplectic fillings, flexibility-rigidity phenomena, and the construction of exotic manifolds.

Future Directions

This work opens several lines for further investigation:

• Extension to higher-codimension Legendrians: Adapting the construction to codimension-1 and codimension-2 Legendrians in contact $(M^5,\xi)$4-manifolds.
• Relation to holomorphic curve invariants: Exploring connections between maximal page crossing numbers and pseudo-holomorphic curve counts, possibly offering new computational techniques for invariants in symplectic field theory.
• Expanding the class of admissible open books: Investigating whether similar invariants can be constructed in cases where the open book has less restrictive geometric constraints (e.g., non-Weinstein, non-simply-connected pages).

Conclusion

The maximal page crossing number invariants defined in this paper offer a new, computable, and isotopy-sensitive Legendrian invariant for surfaces in closed contact 5-manifolds supporting admissible open book decompositions. These invariants fill a critical gap left by the Thurston-Bennequin invariant in higher dimensions, enable practical Legendrian classification, and showcase the utility of open book decompositions in contact and symplectic topology. Their development points to richer invariants and computational techniques for Legendrian submanifolds beyond classical invariants.