Legendrian grid number one knots and augmentations of their differential algebras

TL;DR

This paper investigates the DGA invariants of Legendrian knots in tight lens spaces, providing a method to compute these invariants from grid diagrams and exploring how augmentations depend on the lens space parameters.

Contribution

It introduces a construction for computing DGA invariants from grid diagrams in lens spaces and analyzes how augmentations relate to the space's parameters.

Findings

Augmentations depend solely on p for certain knot families in L(p, p-1).

A method to construct Lagrangian diagrams from grid diagrams is developed.

The study advances understanding of Legendrian knot invariants in lens spaces.

Abstract

In this article we study the differential graded algebra (DGA) invariant associated to Legendrian knots in tight lens spaces. Given a grid number one diagram for a knot in L(p, q), we show how to construct a special Lagrangian diagram suitable for computing the DGA invariant for the Legendrian knot specified by the diagram. We then specialize to L(p, p - 1) and show that for two families of knots, the existence of an augmentation of the DGA depends solely on the value of p.