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On knot types of clean Lagrangian intersections in $T^*\mathbb{R}^3$

Yukihiro Okamoto

TL;DR

The paper develops a framework linking clean Lagrangian intersections in $T^*\mathbb{R}^3$ to knot invariants via Chekanov-Eliashberg DGAs and augmentation varieties. By constructing exact Lagrangian cobordisms and DGA maps between Legendrian ends, it translates geometric intersection data into algebraic constraints on knot types. The main results include rigidity phenomena: if the initial knot $K_0$ is the unknot, the intersecting knot $K$ must also be the unknot, and similar, more nuanced constraints hold for trefoils and torus knots. These results are grounded in the knot DGA, fragmented into a framed knot DGA, and exploit augmentation varieties to compare knot types under Hamiltonian isotopies of conormal bundles. The approach highlights how SFT and Floer-theoretic methods in cotangent bundles yield concrete topological consequences for knot theory through DGA maps and augmentation data.

Abstract

Let $K_0$ and $K$ be knots in $\mathbb{R}^3$. Suppose that by a compactly supported Hamiltonian isotopy on $T^*\mathbb{R}^3$, the conormal bundle of $K_0$ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along $K$. In this paper, we prove some constraints on the pair of knot types of $K_0$ and $K$. One example is that if $K_0$ is the unknot, then $K$ is also the unknot. We also consider some cases where $K_0$ and $K$ have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of $K_0$ and $K$ determined by these DGAs.

On knot types of clean Lagrangian intersections in $T^*\mathbb{R}^3$

TL;DR

The paper develops a framework linking clean Lagrangian intersections in to knot invariants via Chekanov-Eliashberg DGAs and augmentation varieties. By constructing exact Lagrangian cobordisms and DGA maps between Legendrian ends, it translates geometric intersection data into algebraic constraints on knot types. The main results include rigidity phenomena: if the initial knot is the unknot, the intersecting knot must also be the unknot, and similar, more nuanced constraints hold for trefoils and torus knots. These results are grounded in the knot DGA, fragmented into a framed knot DGA, and exploit augmentation varieties to compare knot types under Hamiltonian isotopies of conormal bundles. The approach highlights how SFT and Floer-theoretic methods in cotangent bundles yield concrete topological consequences for knot theory through DGA maps and augmentation data.

Abstract

Let and be knots in . Suppose that by a compactly supported Hamiltonian isotopy on , the conormal bundle of is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along . In this paper, we prove some constraints on the pair of knot types of and . One example is that if is the unknot, then is also the unknot. We also consider some cases where and have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of and determined by these DGAs.
Paper Structure (30 sections, 27 theorems, 174 equations)

This paper contains 30 sections, 27 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Clean intersection in the cotangent bundle
  3. Preliminaries
  4. Lagrangian cobordism arising from Lagrangian submanifolds with clean intersection
  5. Floer cohomology of Lagrangian submanifolds with clean intersection
  6. Gradings and orientations of Lagrangian submanifolds in $T^*Q$
  7. Definition of Floer cohomology of graded Lagrangian submanifolds
  8. Floer cohomology of clean Lagrangian intersections and the triangle product
  9. SFT for Lagrangian cobordisms
  10. Setup
  11. Pseudo-holomorphic curves with boundary in an exact Lagrangian cobordism
  12. Chekanov-Eliashberg DGA over $\mathbb{Z}[H_1(\Lambda)]$
  13. DGA map for Lagrangian cobordism
  14. Application to clean Lagrangian intersection in $T^*\mathbb{R}^{n}$
  15. $1$-dimensional clean intersection in $T^*\mathbb{R}^3$
  16. ...and 15 more sections

Key Result

Theorem 1.1

There exist $a,b\in \mathbb{Z}$ with $a \equiv 1 \mod 2$ such that the set is a subset of $V_{K_0}$.

Theorems & Definitions (66)

  • Theorem 1.1: Theorem \ref{['cor-aug']}
  • Remark 1.2
  • Theorem 1.3: Proposition \ref{['cor-unknot']}
  • Theorem 1.4: Proposition \ref{['prop-trefoil']}
  • Theorem 1.5: Proposition \ref{['prop-torus']}
  • Theorem 1.6: Proposition 6.29 of GP
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 56 more