# Legendrian Rack Invariants of Legendrian Knots

## TL;DR

This paper introduces Legendrian racks, a new algebraic structure inspired by Legendrian knot moves, and demonstrates their effectiveness in distinguishing Legendrian knots beyond smooth knot equivalence.

## Contribution

The paper defines Legendrian racks, provides explicit examples, classifies small cases, and shows their utility in differentiating Legendrian knots.

## Key findings

- Legendrian racks can distinguish certain Legendrian knots.
- Explicit examples of Legendrian racks are provided.
- Classification of Legendrian structures on small racks is achieved.

## Abstract

We define a new algebraic structure called Legendrian racks or racks with Legendrian structure, motivated by the front-projection Reidemeister moves for Legendrian knots. We provide examples of Legendrian racks and use these algebraic structures to define invariants of Legendrian knots with explicit computational examples. We classify Legendrian structures on racks with 3 and 4 elements. We use Legendrian racks to distinguish certain Legendrian knots which are equivalent as smooth knots.