The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant

TL;DR

This paper investigates the contact homology of Legendrian knots with maximal Thurston-Bennequin invariant, revealing examples with both trivial and nonvanishing contact homology despite identical classical invariants.

Contribution

It demonstrates the existence of Legendrian knots with maximal Thurston-Bennequin invariant that have trivial contact homology, contrasting with knots of the same type having nonvanishing contact homology.

Findings

Existence of Legendrian knots with trivial contact homology

Existence of Legendrian knots with nonvanishing contact homology

Knots with identical classical invariants can have different contact homology

Abstract

We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.