Invariants for Legendrian knots in lens spaces

TL;DR

This paper introduces new algebraic invariants for primitive Legendrian knots in lens spaces, extending Legendrian knot theory by defining differential graded algebra invariants tailored for these 3-manifolds.

Contribution

It develops two novel invariants, including a DGA similar to Sabloff's and an enhanced DGA with cyclic group action, specifically for Legendrian knots in lens spaces.

Findings

Defined a DGA invariant from Lagrangian projections

Extended the invariant to include cyclic group actions

Provided methods to compute invariants from p-fold covers

Abstract

In this paper we define invariants for primitive Legendrian knots in lens spaces L(p,q) for q not equal to 1. The main invariant is a differential graded algebra which is computed from a labeled Lagrangian projection of the pair (L(p,q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S^1-bundles over Riemann surfaces. The second invariant defined for knots in lens spaces takes the form of a DGA enhanced with a free cyclic group action and can be computed from the p-fold cover of the pair (L(p,q), K).