An obstruction to decomposable exact Lagrangian fillings

TL;DR

This paper investigates properties of decomposable exact Lagrangian fillings of Legendrian links, establishing a parity condition on associated normal rulings and providing examples of Legendrian knots without such fillings due to ruling constraints.

Contribution

It proves that normal rulings associated with decomposable exact Lagrangian fillings must have an even number of clasps and identifies specific Legendrian knots lacking these fillings.

Findings

Normal rulings linked to fillings have even clasps

Certain Legendrian torus knots lack decomposable fillings

Normal ruling parity constrains fillability

Abstract

We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$, we may obtain a normal ruling of $K$ associated with $L$. We prove that the associated normal rulings must have even number of clasps. As a result, we give a particular Legendrian $(4,-(2n+5))$-torus knot, for each $n \geq 0$, which does not have a decomposable exact Lagrangian filling because it has only 1 normal ruling and this normal ruling has odd number of clasps.