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Non-decomposable Lagrangian cobordisms between Legendrian knots

Roman Golovko, Daniel Komárek

TL;DR

The paper constructs a broad family of non-decomposable Lagrangian cobordisms between stabilized Legendrian knots in the standard contact $S^3$ by leveraging Livingston’s obstructions on the existence of ribbon cobordisms. The core method starts from a decomposable Lagrangian concordance from the tb$=-1$ unknot $U$ to a Legendrian knot $igl\Lambdaigr)$ with an odd prime divisor $p$ of $\det(\Lambda)$, builds stabilized connected sums $\Lambda^n$ and $U^n$, converts the resulting concordance to a Lagrangian one via $h$-principle and Rizell’s approximation, and then assembles genus-$g$ cobordisms $L_g$ using genus-one pieces from Sabloff–Vela-Vick–Wong. The key obstruction comes from Livingston’s estimates together with Borodzik–Truöl, which force at least $c_2(L_g) \ge n/2 - g$ index-2 critical points, and by picking $n>2g$ the cobordism becomes non-decomposable. The paper also provides applicability examples with pretzel knots and ribbon cobordisms, and discusses limitations and potential extensions to fillable ends. This advances understanding of non-decomposable Lagrangian cobordisms and their role in the partial order of Legendrian knots.

Abstract

For a given $g>0$, we construct a family of non-decomposable Lagrangian cobordisms of genus $g$ between (stabilized) Legendrian knots in the standard contact three-sphere. The main technique we use to obstruct decomposability is based on Livingston's estimates.

Non-decomposable Lagrangian cobordisms between Legendrian knots

TL;DR

The paper constructs a broad family of non-decomposable Lagrangian cobordisms between stabilized Legendrian knots in the standard contact by leveraging Livingston’s obstructions on the existence of ribbon cobordisms. The core method starts from a decomposable Lagrangian concordance from the tb unknot to a Legendrian knot with an odd prime divisor of , builds stabilized connected sums and , converts the resulting concordance to a Lagrangian one via -principle and Rizell’s approximation, and then assembles genus- cobordisms using genus-one pieces from Sabloff–Vela-Vick–Wong. The key obstruction comes from Livingston’s estimates together with Borodzik–Truöl, which force at least index-2 critical points, and by picking the cobordism becomes non-decomposable. The paper also provides applicability examples with pretzel knots and ribbon cobordisms, and discusses limitations and potential extensions to fillable ends. This advances understanding of non-decomposable Lagrangian cobordisms and their role in the partial order of Legendrian knots.

Abstract

For a given , we construct a family of non-decomposable Lagrangian cobordisms of genus between (stabilized) Legendrian knots in the standard contact three-sphere. The main technique we use to obstruct decomposability is based on Livingston's estimates.

Paper Structure

This paper contains 5 sections, 3 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Applicability of Theorem \ref{['mainthgenusg']}
  3. Proof of Theorem \ref{['mainthgenusg']}
  4. Lagrangian cobordisms constructed by Sabloff, Vela-Vick and Wong
  5. Livingston's estimates and the results of Borodzik--Truöl

Key Result

Theorem 1

Let $\Lambda$ be a Legendrian knots in $(S^3, \xi_{st})$ such that Then for a given $g>0$ there is a pair of Legendrian knots $(\Lambda_-, \Lambda_+)$, where $\Lambda_-$ is a sufficiently many times stabilized version of $\Lambda^{n(g)}=\Lambda\#\dots \#\Lambda$ and $\Lambda_+$ is a sufficiently many times stabilized version of $U^{n(g)}=U\# \dots \#U$, and a non-d

Figures (3)

  • Figure 1: Left: Lagrangian filling of the $tb=-1$ unknot; Right: Lagrangian pair-of-pants cobordism obtained from the Lagrangian $1$-handle attachment.
  • Figure 2: Decomposable Lagrangian concordance $C_{k}$ from the $tb=-1$ Legendrian unknot $U$ to the the Legendrian representative $\Lambda_{k}$ of the pretzel knot $P(3,-3,k)$ induced by the ambient surgery along the red arc. The front projection of $\Lambda_{k}$ has $k-3$ crossings in the blue box.
  • Figure 3: Lagrangian cobordism of genus one from $(S_+S_-)(\Lambda')$ to $\Lambda'$ obtained by making an ambient surgery along two red arcs.

Theorems & Definitions (6)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 5: CritpointsknotcobordismsBorodzikTruol25
  • Proposition 6: BorodzikTruol25