Legendrian torus knots in lens spaces

TL;DR

This paper classifies topological and Legendrian torus knots in lens spaces, providing a comprehensive understanding of their types and Legendrian representatives within tight contact structures.

Contribution

It offers the first classification of all topological torus knots on Heegaard tori in lens spaces and classifies Legendrian representatives up to contactomorphism.

Findings

Complete classification of topological torus knots in lens spaces.

Classification of positive Legendrian torus knots in tight contact structures.

Results applicable to contact topology and knot theory in 3-manifolds.

Abstract

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.