On invariants for Legendrian knots

TL;DR

This paper explores the relationship between sutured invariants and Legendrian invariants of knots in contact 3-manifolds, establishing new vanishing results and reinterpreting known properties through this connection.

Contribution

It links the sutured invariant of a knot complement to the Legendrian invariant, providing a new perspective and a vanishing theorem related to Giroux torsion.

Findings

Derived a vanishing theorem for Legendrian invariants with Giroux torsion

Connected sutured invariants to Legendrian invariants

Reproved known properties of Legendrian invariants

Abstract

Suppose that L is a null--homologous Legendrian knot in a contact 3--manifold. We determine the connection between the sutured invariant of the complement of L and the Legendrian invariant defined by Lisca, Ozsvath, Stipsicz and Szabo. In particular, we derive a vanishing theorem for the Legendrian invariant in the presence of Giroux torsion in the complement of the knot, and reprove several known properties of the Legendrian invariant from this perspective.