# Categorification of Legendrian knots

## TL;DR

This paper explores a real analogue of perverse schobers, categorifying Legendrian knots and points, and connecting various advanced concepts like semi-orthogonal decompositions and spherical functors.

## Contribution

It introduces a novel categorification framework for Legendrian knots, extending the concept of perverse schobers to a real setting with multiple related notions.

## Key findings

- Connections between Legendrian knots and advanced categorical concepts
- Introduction of a real analogue of perverse schobers
- Framework unifies various notions like mutations and spherical functors

## Abstract

Perverse schober defined by Kapranov--Schechtman is a categorification of the notion of perverse sheaf. In their definition, a key ingredient is certain purity property of perverse sheaves. In this short note, we attempt to describe a real analogue of the above story, as categorification of Legendrian points/knots. The notion turns out to include various notions such as semi-orthogonal decomposition, mutation braiding, spherical functor, N-spherical functor, and irregular perverse schober.