On loose Legendrian knots in rational homology spheres

TL;DR

This paper demonstrates that loose Legendrian knots in certain 3-manifolds are classified by classical invariants under specific conditions, utilizing key results from Dymara and Eliashberg.

Contribution

It establishes a classification criterion for loose Legendrian knots in rational homology spheres based on classical invariants.

Findings

Loose Legendrian knots with same invariants are isotopic under certain conditions

Utilizes Dymara's result on loose Legendrian knots

Applies Eliashberg's classification of overtwisted contact structures

Abstract

We prove that loose Legendrian knots in a rational homology contact 3-sphere, satisfying some additional hypothesis, are Legendrian isotopic if and only if they have the same classical invariants. The proof requires a result of Dymara on loose Legendrian knots and Eliashberg's classification of overtwisted contact structures on 3-manifolds.