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Double-point enhanced GRID invariants and Lagrangian cobordisms

Ashton Lewis, Zachary Ojakli, Ina Petkova, Benjamin Shapiro

TL;DR

The work introduces double-point enhanced GRID invariants for Legendrian links by defining canonical GRID-state classes in double-point enhanced grid homology and proves their invariance under grid commutations and stabilizations. It then establishes a weak functorial framework for these invariants under decomposable Lagrangian cobordisms via pinch and birth moves, producing obstructions to the existence of such cobordisms and to decomposable fillings. The paper also compares the new invariants to the classical GRID invariants, providing computations and evidence that the enhanced and standard invariants align in practice, while outlining open questions about equivalence in full generality. Overall, the results offer a new, robust set of Legendrian/transverse invariants that obstruct decomposable cobordisms and enrich the grid-homology toolkit for low-dimensional contact geometry.

Abstract

We define an invariant of Legendrian links in the double-point enhanced grid homology of a link, and prove that it obstructs decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb R^3$.

Double-point enhanced GRID invariants and Lagrangian cobordisms

TL;DR

The work introduces double-point enhanced GRID invariants for Legendrian links by defining canonical GRID-state classes in double-point enhanced grid homology and proves their invariance under grid commutations and stabilizations. It then establishes a weak functorial framework for these invariants under decomposable Lagrangian cobordisms via pinch and birth moves, producing obstructions to the existence of such cobordisms and to decomposable fillings. The paper also compares the new invariants to the classical GRID invariants, providing computations and evidence that the enhanced and standard invariants align in practice, while outlining open questions about equivalence in full generality. Overall, the results offer a new, robust set of Legendrian/transverse invariants that obstruct decomposable cobordisms and enrich the grid-homology toolkit for low-dimensional contact geometry.

Abstract

We define an invariant of Legendrian links in the double-point enhanced grid homology of a link, and prove that it obstructs decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on .
Paper Structure (18 sections, 15 theorems, 79 equations, 13 figures)

This paper contains 18 sections, 15 theorems, 79 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The double-point enhanced GRID invariants
  3. Obstructions to decomposable Lagrangian cobordisms
  4. Comparison with the GRID invariants
  5. Organization
  6. Preliminaries
  7. Legendrian Knots and Lagrangian cobordisms
  8. Knot Floer homology, the GRID invariants, and double points
  9. The double-point enhanced GRID invariants
  10. Definition of the double-point enhanced GRID invariants
  11. Invariance under commutations
  12. Invariance under stabilization and destabilization
  13. The proof of invariance completed
  14. The double-point enhanced GRID invariant: Obstructing decomposable Lagrangian cobordisms
  15. Pinches
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $\mathbb{G}$ and $\mathbb{G}'$ are two grid diagrams that represent the same Legendrian link $\Lambda \subset (S^3, \xi_{\mathrm{std}})$. Then, there exists a bigraded isomorphism with $\Phi(\lambda^+_{\mathit{big}}(\mathbb{G})) = \lambda^+_{\mathit{big}}(\mathbb{G}')$ and $\Phi(\lambda^-_{\mathit{big}}(\mathbb{G})) = \lambda^-_{\mathit{big}}(\mathbb{G}')$.

Figures (13)

  • Figure 1: An example of a front projection.
  • Figure 2: The moves on front projections that correspond to elementary cobordisms; vertical and horizontal reflections of these moves are also allowed. The first three moves are Legendrian Reidemeister moves, the fourth move is called a pinch, and the fifth move is called a birth.
  • Figure 3: The combined diagram of a row commutation involving $\alpha$ and $\alpha'$.
  • Figure 4: An example of a domain with two decompositions into a rectangle and a pentagon, where exactly one of the pentagons is long.
  • Figure 5: Left: an $X$-marked square in $\mathbb{G}$. The distingushed 2x2 square of the diagram $\mathbb{G}'$ obtained from $\mathbb{G}$ by stabilization at this $X$ marking, where the stabilization is of type X:SE, X:NW, X:SW, and X:NE, as seen from left to right.
  • ...and 8 more figures

Theorems & Definitions (37)

  • Theorem 1.1
  • Corollary 1.1
  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.2
  • Theorem 1.4
  • Corollary 1.3
  • Theorem 1.5
  • Theorem 1.6
  • ...and 27 more