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Khovanov homology and Lagrangian cobordisms

Gage Martin, Ina Petkova, Zachary Winkeler

TL;DR

The paper investigates how Khovanov homology detects decomposable Lagrangian cobordisms between Legendrian links, focusing on decomposable Lagrangian fillings. By deforming such cobordisms to transverse ascending cobordisms, it leverages the functoriality of Khovanov maps to show F_L sends Plamenevskaya's invariant ψ(K+) to ±ψ(K−), yielding obstructions and tying the invariants to Ng's line. It derives a topological tb-bound and Ng-line constraints for links with decomposable fillings, and introduces filtered invariants from the Lee deformation that remain functorial under decomposable cobordisms, offering potentially stronger obstructions in cases where ψ is not definitive. Together, these results connect Khovanov theory with contact-topological questions about cobordisms and provide new computable obstructions to decomposable Lagrangian cobordisms.

Abstract

We provide a partial answer to a question of Ekholm, Honda, and Kálmán about the relationship between Khovanov homology and decomposable Lagrangian cobordisms. We also utilize previously defined filtered invariants to give obstructions to decomposable Lagrangian cobordisms from Khovanov homology.

Khovanov homology and Lagrangian cobordisms

TL;DR

The paper investigates how Khovanov homology detects decomposable Lagrangian cobordisms between Legendrian links, focusing on decomposable Lagrangian fillings. By deforming such cobordisms to transverse ascending cobordisms, it leverages the functoriality of Khovanov maps to show F_L sends Plamenevskaya's invariant ψ(K+) to ±ψ(K−), yielding obstructions and tying the invariants to Ng's line. It derives a topological tb-bound and Ng-line constraints for links with decomposable fillings, and introduces filtered invariants from the Lee deformation that remain functorial under decomposable cobordisms, offering potentially stronger obstructions in cases where ψ is not definitive. Together, these results connect Khovanov theory with contact-topological questions about cobordisms and provide new computable obstructions to decomposable Lagrangian cobordisms.

Abstract

We provide a partial answer to a question of Ekholm, Honda, and Kálmán about the relationship between Khovanov homology and decomposable Lagrangian cobordisms. We also utilize previously defined filtered invariants to give obstructions to decomposable Lagrangian cobordisms from Khovanov homology.

Paper Structure

This paper contains 3 sections, 10 theorems, 15 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Decomposable Lagrangian cobordisms and Plamanevskaya's invariant
  3. Filtered invariants and decomposable Lagrangian cobordisms

Key Result

Theorem 1.1

Let $\Lambda_-$ and $\Lambda_+$ be two Legendrian links with positive transverse pushoffs $K_-$ and $K_+$, and suppose there is a decomposable Lagrangian cobordism $L$ from $\Lambda_-$ to $\Lambda_+$. Then the induced cobordism map $F_L: \mathop{\mathrm{Kh}}\nolimits(\Lambda_+)\to \mathop{\mathrm{K

Figures (1)

  • Figure 1: Left: An elementary Lagrangian cobordism for a 1-handle attachment (a pinch) from $\Lambda_-$ to $\Lambda_+$. Right: The corresponding ascending cobordism (a band attachment followed by an isotopy) between the positive transverse pushoffs of $\Lambda_-$ and $\Lambda_+$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • proof : Proof of \ref{['thm:KhLagrangian']}
  • ...and 5 more