# Arboreal singularities and loose Legendrians I

## TL;DR

This paper explores the relationship between arboreal singularities and loose Legendrians, demonstrating that constructable sheaves can detect looseness in arboreal links, thus advancing understanding of their geometric and topological properties.

## Contribution

It establishes that for linear arboreal singularities, constructable sheaves are sufficient to determine looseness of closed sets in arboreal links.

## Key findings

- Constructable sheaves detect looseness in arboreal links.
- Linear arboreal singularities are characterized by their links in Legendrian spaces.
- The work bridges Lagrangian singularities with Legendrian topology.

## Abstract

Arboreal singularities are an important class of Lagrangian singularities. They are conical, meaning that they can be understood by studying their links, which are singular Legendrian spaces in $S^{2n-1}_{\text{std}}$. Loose Legendrians are a class of Legendrian spaces which satisfy an $h$--principle, meaning that their geometric classification is in bijective correspondence with their topological types. For the particular case of the linear arboreal singularities, we show that constructable sheaves suffice to detect whether any closed set of an arboreal link is loose.